Sequences and series. describing sequences

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1 5 5A Describing sequences 5B Arithmetic sequences 5C Arithmetic series 5D Geometric sequences 5E Geometric series 5F Applications of sequences and series Sequences and series areas of STudY Sequences and series as maps between the natural numbers and real numbers, and the use of technology to generate sequences and series and their graphs Sequences generated by recursion: arithmetic (t n + = t n + d), geometric (t n + = rt n ) and fixed point iteration (for example, t =, t n + = t n ; t = 5, t n + = 0.8t n ( t n )) Practical applications of sequences and series, such as financial arithmetic and population modelling. ebookplus 5a describing sequences Digital doc 0 Quick Questions Sequences of numbers play an important part in our everyday life. For example, the following sequence:.5,.37,.58,.57,.63,... gives the end-of-day trading price (for 5 consecutive days) of a share in an electronics company. It looks like the price is on the rise, but is it possible to accurately predict the future price per share of the com pany? The following sequence is more predictable: 0 000, 9000, 800,... This is the estimated number of radioactive decays of a medical compound each minute after adminis tration to a patient. The compound is used to diagnose tumours. In the first minute, radioactive decays are predicted; during the second minute, 9000, and so on. Can you predict the next number in the sequence? You re correct if you said 790. Each suc cessive term here is 90% of, or 0.90 times, the previous term. maths Quest Standard General mathematics for the Casio Classpad

2 Sequences are strings of numbers. They may be finite in number or infinite. Number sequences may follow an easily recognisable pattern or they may not. A great deal of recent mathematical work has gone into deciding whether certain strings follow a pattern (in which case subsequent terms could be predicted) or whether they are random (in which case subsequent terms cannot be predicted). This work forms the basis of chaos theory, speech recognition software for com puters, weather prediction and stock market fore casting, to name but a few uses. The list is almost endless. Sequences which follow a pattern can be described a number of different ways. They may be listed in sequential order, they may be described as a func tional definition, or they may be described in an iterative definition. Listing in sequential order Consider the sequence of numbers t : {5, 7, 9,...}. The numbers in sequential order are firstly 5 then 7 and 9 with the indication that there are more numbers to follow. The symbol t is the name of the sequence and the first three terms in the sequence shown are t = 5, t = 7 and t 3 = 9. The fourth term, t 4 if the pattern were to continue, would be the number. In general, t n is the nth term in the sequence. In this example, the next term is simply the previous term with the number added to it, with the first term being the number 5. Another possible sequence is t : {5, 0, 0, 40,...}. In this case it appears that the next term is twice the previous term. The fifth term here, if the pattern continued, would be t 5 = 80. It can be difficult to determine whether or not a pattern exists in some sequences. Can you find the next term in the following sequence? t : {,,, 3, 5, 8,...} Here the next term is the sum of the previous two terms, hence the next term would be which is equal to 3, and so on. This sequence is called the Fibonacci sequence and is named after its discoverer, Leon ardo Fibonacci, a thirteenth century mathematician. Here is another sequence; can you find the next term here? t : {7,, 6,, 9,...} In this sequence the difference between successive terms increases by for each pair. The first difference is 4, the next difference is 5 and so on. The sixth term is thus 37 which is 8 more than 9. Functional definition A functional definition is expressed in the form: t n = n - 7, n {,, 3, 4,...} Using this definition the nth term can be readily calculated. For this example t = - 7 = - 5, t = - 7 = - 3, t 3 = 3-7 = - and so on. We can readily calculate the 00th term, t 00 = 00-7 = 93, simply by substituting the value n = 00 into the expression for t n. Look at the following example: d n = 4.9n, n {,, 3,...} For this example, in which the sequence is given the name d, d = 4.9 = 4.9, d = 4.9 = 9.6. Listing the sequence would yield d : {4.9, 9.6, 44., 78.4,...}. The 0th term would be = 490. Here is another example: c n = cos (np) +, n {,, 3,...} Here the sequence would be c: {0,, 0,,...}. 3 Iterative definition An iterative definition is expressed in the form: t n + = 3t n - ; t = 6 Chapter 5 Sequences and series 3

3 This definition looks complicated, but is actually straightforward. You may have already come across this idea on a spreadsheet. The word iter ation means the calculation of the next term from the previous term using the same procedure. The symbol t n + simply means the next term after the term t n. In the above example the first term, t, is 6 (this is given in the definition) and so the next term, t, is = 6, and the following term is = 46. In each and all cases the next term is found by multiplying the previous term by 3 and then sub tracting. We could write the sequence out as a table: n t n Comment t = 6 t = 3t - = = 6 t 3 = 3t - = = 46 3 t 4 = 3t 3 - = = 36 Given in the definition Using t to find the next term, t Using t to find the next term, t 3 Using t 3 to find the next term, t 4 An example of this sequence using notation found in a spreadsheet would be: A = 6 (the first term is equal to 6) A = 3 A - (the next term is 3 times the previous term minus ). You could then apply the Fill Down option in the Edit menu of the spreadsheet from cell A downwards to generate as many terms in the sequence as required. This would result in the next cell down being three times the previous cell, less. The iterative definition finds a natural use in a spreadsheet environment and consequently much use is made of it. A drawback is that you cannot find the nth term directly as in the functional definition, but the advantage is that more complicated systems can be successfully modelled using iterative descriptions and hence are more interesting and relevant. Worked example a Find the next three terms in the sequence, b: {4, 7, 7,...}. b Find the 4th, 8th and th term in the following sequence: e n = n - 3n, n {,, 3,...}. c Find the nd, 3rd and 5th term for the following sequence: k n + = k n +, k = ebookplus Tutorial int-0993 Worked example a ThInk In this example the sequence is listed and a simple pattern is evident. From inspection, the next term is half the previous term and so the sequence would be 4, 7, 7, 7 4, 7 8, 7 6. b This is an example of a functional definition. The nth term of the sequence is found simply by substitution into the expression e n = n - 3n. WrITe a The next three terms are 7 4, 7 8, 7 6. b e n = n - 3n 4 maths Quest Standard General mathematics for the Casio Classpad

4 First define the rule as follows. On the Main screen, tap: Action Command Define Complete the entry line as: Define e(n) = n - 3n Then press E. Find all required terms. For instance, to find the 4th term, complete the entry line as: e(4) Then press E. 3 Write the solution. e 4 = 4; e 8 = 40; e = 08. c This is an example of an iterative definition. We can find the nd, 3rd and 5th terms for the sequence k n + = k n +, k = by iteration. c k n + = k n +, k = Substitute k = into the formula to find k. k = = 0 3 Continue the process until the value of k 5 is found. k 3 = 0 + = k 4 = + = 3 k 5 = 3 + = 7 4 Write the answer. Thus k = 0, k 3 = and k 5 = 7. 5 Alternatively, on the Sequence screen, tap &. Complete the entry line as shown at right. Note: Use the V tab at the top of the screen to type the sequence. 6 Tap #. The terms of the sequence are now in column a n and can be read directly from the screen. 7 Write the answer from the screen. Chapter 5 Sequences and series 5

5 Logistic equation The logistic equation is a model of population growth. It gives the rule for determining the population in any year, based on the population in the previous year. Since we need the previous term in order to be able to generate the next term of the sequence, the logistic equation is an example of an iterative definition. It is of the general form: t n + = at n ( - t n ), where 0 < t 0 < and a is a constant. Depending on the value of a, sequences generated by use of the logistic equation could be convergent, divergent, or oscillating. A string of numbers that converges to (settles at) a certain fixed value is called a convergent sequence. Sequence t n can converge to only one possible number, x, called the limit of the sequence. This can be written as t n x. (The symbol is read as tends to, or approaches.) A sequence whose terms grow further and further apart is called divergent. That is, a sequence is divergent if t n, or t n - as n. Finally, a sequence whose terms tend to fluctuate between two (or more) values is called oscillating. An oscillating sequence is neither convergent nor divergent. Worked Example Given that a = and t 0 = 0.7, use the logistic equation to generate a sequence of 6 terms, and state whether the sequence is convergent, divergent, or oscillating. If the sequence is convergent, state its limit. Think Write the logistic equation, replacing a with its given value (that is, ). To find t, substitute the value of t 0 (that is, 0.7) in place of t n and evaluate. 3 To find the next term, t, substitute the value of t (that is, 0.4) in place of t n and evaluate. 4 Continue the iterative process four more times, each time substituting the value of the previous term into the logistic equation to find the next term. 5 The terms of the sequence are growing closer and closer to each other, finally settling at 0.5. Write t n + = at n ( t n ) = t n ( t n ) t = t 0 ( t 0 ) = 0.7 ( 0.7) = 0.4 t = t ( t ) = 0.4 ( 0.4) = t 3 = t ( t ) = ( 0.487) = t 4 = t 3 ( t 3 ) = ( ) = t 5 = t 4 ( t 4 ) = ( ) = 0.5 t 6 = t 5 ( t 5 ) = 0.5 ( 0.5) = 0.5 The sequence is convergent; the limit of the sequence is 0.5. Note that instead of saying the limit of the sequence is 0.5 in the pre vious example, we could simply write t n Maths Quest Standard General Mathematics for the Casio ClassPad

6 REMEMBER. A sequence is a string of numbers or expressions which may follow a recognisable pattern.. A sequence can be described in a number of ways. (a) As a list for example: t n : {, 7,, 6,...} (Note: t 3 = ) (b) As a function for example: t n = n - n, n {,, 3,...} (Note: t 5 = 5-5 = - 5) (c) As a recursive or iterative formula for example: t n + = t n - 3, t = 6 (Note: t = 6-3 = 9) 3. The logistic equation is a model of population growth and is an example of an iterative definition. It is of the general form: t n + = at n ( t n ), where 0 < t 0 < and a is a constant. 4. A sequence that converges to (settles at) a certain fixed number, x (the limit of the sequence) is called convergent. This can be written as t n x. A sequence whose terms grow further and further apart is called divergent. That is, a sequence is divergent if t n, or t n - as n. A sequence whose terms fluctuate between two (or more) values is called oscillating. An oscillating sequence is neither convergent nor divergent. Exercise 5A Describing sequences WEa For each of the following sequences, find the next three terms. a {, 4, 7, } b {, 0, -, -, } c {, 4, 6, 64, } d {3, 3, 3, } 4 e {, - 5, 8, -, 4, } g {3, 4, 7,, 8, } i {, 0, -, 0,, } k {04, - 5, 56, - 8, } f {, 5, 9, 4, 0, } h {a - 5b, a - b, b, - a + 4b, } j {.0,.,., } WEb Find the first, fifth and tenth terms in the following sequences. a t n = n - 5, n {,, 3, } b t n = 4 3 n -, n {,, 3, } n c t n =, n {,, 3, } n + d t n = 7-3.7n, n {,, 3, } e t n = 5 n, n {,, 3, } f tn = 5 ( 3 n ), n {,, 3, } g t n = ( - ) n + n, n {,, 3, } i t n = n - n + 4, n {,, 3, } k t n = ar n -, n {,, 3, } h t n = 3 n -n, n {,, 3, } j t n = a + (n - )d, n {,, 3, } 3 WEc Using a spreadsheet, or other method, find the third, eighth and tenth terms in the following sequences. a u n + = u n +, u = 3 b u n + = u n -, u = c u n + = 3u n, u = 0.85 d u n + = - u n, u = - 3 e u n + = 3 u 4 n, u = 4 3 f u n + = u n - 7, u = 4 g u n + = - u n +, u = 3 h u n + = u n + ( - ) n u n, u = 3 i u n = u n -, u = 4 j u n + = au n + a, u = a k u n + = u n + + u n, u =, u = l u n + = - u n -, u = 3 Chapter 5 Sequences and series 7

7 4 WE Given the following values of a and t 0, use the logistic equation to generate a sequence of 6 terms. State whether the sequence is convergent, divergent, or oscillating. If the sequence is convergent, state its limit. a a = 0.8, t 0 = 0.5 b a = 0.4, t 0 = 0.6 c a =., t 0 = 0.9 d a =.9, t 0 = 0.4 e a =., t 0 = 0.5 f a =.5, t 0 = 0.3 g a = 3, t 0 = 0. h a = 3.4, t 0 = 0.7 i a = 4., t 0 = 0. j a = 4.5, t 0 = Study the pattern in each of the following sequences and where possible write the next two terms in the sequence, describing the pattern that you use. a 5, 6, 8,, b 4, 9,, 3,, 9, c 9, 8, 9, 0, d 6,,, 6,, e 5, 8, 3,, f, 3, 7, 5, g, 3,, 4, 3, 6 MC a Which of the following functional definitions could be used to describe a sequence {3,, -, }? A t n = n -, n {,, 3, } B t n = n - 5, n {,, 3, } C t n = 5n -, n {,, 3, } D t n = 5 - n, n {,, 3, } E t n = (5 - n), n {,, 3, } b Which of the following recursive definitions could be used to describe a sequence {0, 0, 5, }? A t n + = t n - 30, t = 0 B t n + = t n, t = - 0 C t n + = t n - t n, t = 0 D t n + = t n - 0, t = 0 E t n + = t n, t = 0 n c Which of the following sequences is generated by the definition tn 6, n {,, 3, }? A { - 3, 6, 5, } B { - 3, 6, -, } C { - 3, 6,, } D { - 3, 6,, } E { - 3, 6, 8, } 7 The first term in a sequence is. The second term is obtained by multiplying the value of the first term by 3, then adding. The third term is obtained by multiplying the value of the second term by 3, then adding. If this pattern continues, the equation which generates this sequence is: A t n = 3t n + + where t = B t n + = 3t n + where t = C 3t n + = t n + where t = D 3t n = t n + + where t = E t n + + = 3t n where t = 8 A sequence follows the rule w n + = 4w n +, where w n is the nth term and n =,, 3, The value of the second term, w, is 0. The value of the fourth term, w 4, is: A 8 B 0 C 4 D 70 E 00 [ VCAA 004] Exam tip In this question, students were given the equation and the value of the second term of the sequence generated by this equation, and were asked to generate the fourth term. Only 39% of students were able to perform this relatively routine task. While this could have been done with the aid of technology, it was best treated as a pencil and paper exercise. [Assessment report 004] 9 Write the iterative definition for each of the following sequences. a {7, 5, 3,, -, } b {, 6, 3,.5, } c {,.6, 3., } d {,, 56, 8, } e {4, -, 36, } f {, 4, 6, 56, } 8 Maths Quest Standard General Mathematics for the Casio ClassPad

8 0 In the township of Grizabella, the population of stray cats in any given year is given as p n +. This can be calculated using the formula p n + =.3p n ( p n ), where p n is the number of cats (in hundreds) in the preceding year. If in 005 there were 8 stray cats in Grizabella township, calculate: a the expected number of stray cats for 006 and 007 b the limiting number of stray cats that Grizabella township can sus tain. In the neighbouring township of Macavity, the size of the population of stray cats follows the logistic equation p n + = 0.3p n ( p n ), where p n + and p n refer to the popu lation size (in hundreds) in any given year and in the preceding year respectively. It is known that in 005, there were 6 stray cats in the township. By generating and examining the sequence of numbers using the above equation, decide what will happen in the long run to the population size of stray cats in Macavity township. (That is, will the population of cats keep increasing, decreasing, or settle at a particular value?) 5B Arithmetic sequences At a racetrack a new prototype racing car unfortunately develops an oil leak. Each second, a drop of oil hits the road. The driver of the car puts her foot on the accelerator and the car increases speed at a steady rate as it hurtles down the straight. The diagram below shows the pattern of oil drops on the road with the distances between the drops labelled. 0 metres 6 metres 4 metres 8 metres 34 metres The sequence of distances travelled in metres each second is {0, 8, 6, 34, 4, }. The first term in the sequence, t, is 0 and as you can see, each subsequent term is 8 more than the previous term. This type of sequence is given a special name an arithmetic sequence. An arithmetic sequence is a sequence where there is a common difference between any two successive terms. We can list the sequence in a table as in table A. From this table we can see that it is possible to write a functional definition for the sequence in terms of the first term, 0, and the common difference, 8, and thus: t n = 0 + (n - ) 8 = + 8n, n {,, 3, } Chapter 5 Sequences and series 9

9 We can readily get a general formula for the nth term of an arithmetic sequence whose first term is a and whose common difference is d (see table B). Table A Table B n t n t n n t n a + 0 d a + d a + d a + 3 d n 0 + (n - ) 8 = 0 + 8n - 8 = + 8n + 8n n a + (n - ) d = (a - d) + dn In general then: The nth term of an arithmetic sequence is given by t n = a + (n - ) d = (a - d) + nd, n {,, 3,...} where a is the first term and d is the common difference. If we consider three successive terms in an arithmetic sequence, namely x, y and z, then since y - x = the common difference, d, and z - y = d, it follows that: y - x = z - y y = z+ x The middle term of any three consecutive terms in an arithmetic sequence is called an arithmetic mean and is the average of the outer two. z x That is, y = + for any 3 consecutive terms, x, y, z of the arithmetic sequence. Worked Example 3 Show that the following sequences are arithmetic. a 7 4, 8, -, b x - 4x, 3x - 7x, 5x - 0x, Think a To show that a sequence is arithmetic you need to show that the difference between any two successive terms is a constant. Find the difference between the first and the second terms. Write a t - t = = = 3 8 Find the difference between the second and the third terms. t 3 - t = = Compare the differences and draw your conclusion. b Find the difference between the first two terms. t - t = t 3 - t = 3 8 The sequence is arithmetic. b t - t = 3x - 7x - (x - 4x) = x - 3x 0 Maths Quest Standard General Mathematics for the Casio ClassPad

10 Find the difference between the second and the third terms. 3 Compare the differences and draw your conclusion. t 3 - t = 5x - 0x - (3x - 7x) = x - 3x t - t = t 3 - t = x - 3x The sequence is arithmetic. Worked Example 4 State which of the following are arithmetic sequences by finding the difference between successive terms. For those which are arithmetic, find the next term in the sequence, t 4, and consequently find the functional definition for the nth term for the sequence, t n. a t : {4, 9, 5, } b t : { -,, 4, } Think Write a To check that a sequence is arithmetic, see if a common difference exists. a 9-4 = = 6 There is no common difference, as 5 6. Since there is no common difference the sequence is not arithmetic. b To check that a sequence is arithmetic, see if a common difference exists. b - - = = 3 The common difference is 3. The sequence is arithmetic with the common difference d = 3. 3 The next term in the sequence, t 4, can be found by adding 3 to the previous term, t 3. 4 To find the functional definition, write the formula for the nth term of the arithmetic sequence. t 4 = t = = 7 t n = a + (n - ) d = a + nd - d = (a - d ) + nd 5 Identify the values of a and d. a = - and d = 3 6 Substitute a = - and d = 3 into the formula and simplify. t n = ( - - 3) + n 3 t n = 3n - 5 Worked Example 5 Find the missing terms in this arithmetic sequence: {4, a, 55, b, }. Think The first three successive terms are 4, a, 55. Write the rule for the middle term of the three successive terms of an arithmetic sequence. Write x z For x, y, z: y = + Identify the variables. x = 4; y = a; z = 55 3 Substitute the values of x, y and z into the formula in step and evaluate a = = 48 Chapter 5 Sequences and series

11 4 Find the common difference. (The second term is now known.) 5 Find the value of b by adding the common difference to the preceding taerm. d = t - t = 48-4 = 7 b = = 6 6 State your answer. So a = 48, b = 6 Worked example 6 Find the 6th and nth term in an arithmetic sequence with the 4th term 5 and 8th term 37. ThInk WrITe ebookplus Tutorial int-0875 Worked example 6 Write the formula for the nth term of the arithmetic sequence. Substitute n = 4 and t 4 = 5 into the formula and label it equation []. 3 Substitute n = 8 and t 8 = 37 into the formula and label it equation []. 4 Solve the simultaneous equations. On the Main screen, set up the two equations by tapping: ) {N Complete the entry lines as shown at right and press E. t n = a + (n - ) d t 4 : a + 3d = 5 [] t 8 : a + 7d = 37 [] 5 Write the solution. Solving a + 3d = 5 and a + 7d = 37 for a and d gives a = 3, d = 6 To find the nth term of the arithmetic sequence, substitute the values of a and d into the general formula and simplify. 7 To find the 6th term, substitute n = 6 into the formula, established in the previous step and evaluate. Alternatively generate the sequence using a calculation. t n = 3 + (n - ) = 3+ n n 4 t n =, n {,, 3,...} 6 4 If n = 6, t 6 = = 8 maths Quest Standard General mathematics for the Casio Classpad

12 remember. An arithmetic sequence is one where successive terms have a common difference. This common difference is given the symbol d. Thus t n + - t n = d for all values of n. The first term in the sequence is given the symbol a.. If x, y, z are successive terms in an arithmetic sequence then y is called an arithmetic x z mean and is given by y = +. That is, the middle term is the average of the outer two terms. 3. An arithmetic sequence can be written as a, a + d, a + d, and so the nth term t n is: t n = a + (n - )d using the function notation, or t n + = t n + d, t = a using the iterative notation. exercise 5B arithmetic sequences We3 Show that the following sequences are arithmetic. a { -, - 7, -, } b { - 0., 3.48, 7.08, } c { 3 8, 3 8, 9 8, } d {.3, -.7, - 5.7, } e { 5 9, 9, 7 9, } f {8, - 8, - 54, } g {5 3, 7 4, , } h {x + 9, x + 7, 3x + 5, } i {3x - 4x, 5x - x, 7x, } j {3( - x), ( - x), - x, } We4 State which of the following are arithmetic sequences by finding the difference between successive terms. For those which are arithmetic, find the next term in the sequence, t 4, and consequently find the functional definition for the nth term for the sequence, t n. a t n : {3, 5, 7, } b t n : {4, 7,, } c t n : {3, 6,, } d t n : { - 3, 0, 3, } e t n : { -, - 6, - 0, } f t n : { 7, 4, 9 7, } g t n : { 3 4, 3, 3, } h t n: { 3 4, 3, 9 4, } i t n : { 4, 3, 3 4, } j t n: {p + 3, 4p +, 6p -, } 3 Find the term given in brackets for each of the following arithmetic sequences. a {4, 9, 4, }, (t ) b { -, 0,, }, (t 58 ) c { - 7, -, 3, } (t 00 ) d {, -, - 4, } (t 05 ) 4 Find the functional definition for the nth term of the following arithmetic sequences: a where the first term is 5 and the common difference is - 3 ebookplus b where the first term is.5 and the common difference is c where the first term is - 3 and the common difference is 3 d where the first term is x and the common difference is 5x. 5 Find the nth term in the arithmetic sequence where the first term is 6 and the third term is 0. Digital doc Spreadsheet 7 Arithmetic sequences 6 Find the nth term in the arithmetic sequence where the first term is 3 and the third term is 3. Chapter 5 Sequences and series 3

13 5C 7 mc The monthly enrolment of students in social dance class follows an arithmetic sequence. If there were 4 students enrolled in the first month of operation and 6 students in the fourth month, how many students will be enrolled in the fifth month, assuming that the pattern will continue? A 8 B 0 C D 4 E 3 8 We5 Find the missing terms in this arithmetic sequence: {6, m, 7, n} 9 Find the missing terms in the arithmetic sequence below. x - 3y,, - 3x + 5y,, 0 We6 Find the 4th term and nth term in the arithmetic sequence whose first term is 6 and whose 7th term is - 0. If t 0 = 00 and t 5 = 75, find the first term, the common dif ference and hence the nth term for the arithmetic sequence. If t 0 = and t 3 = 3, find the first term, the common difference and hence the nth term for the 4 arithmetic sequence. 3 Insert four evenly spaced numbers between 8 and For the arithmetic sequence {, m, n, 37, }, find the values for m and n. 5 For the following arithmetic sequences, find the iterative definition and use it in a spreadsheet to generate the first 50 numbers in the sequence using the Fill Down option. a t n : {3, 7,, } b t n : { - 3, 0, 3, } c t n : { -, - 6, - 0 } d t n : {,, 9, } e t n : { 3 4, 3, 9 4, } f t n: { 4, 3, 3 4, } g t n : {p + 3, 4p +, 6p -, } 6 The first three terms in an arithmetic sequence are 37, 3, 7 and another term is - 3. Which term is - 3? 7 Find the value of x such that the following forms an arithmetic pro gression: x, 3x + 4, 0x mc For the following sequence t : {4,, 8, }, the difference between the 4th and the 0th term is: A 35 B 4 C 49 D 56 E 63 9 mc The tenth term in an arithmetic sequence is and the third term is -. The first term in the sequence is: A - 7 B - 3 C - 5 D - 8 E The ratio (division) between the first term and the second term in an arithmetic sequence is 3. 4 The ratio between the second term and the third term is 4. 5 a Calculate the ratio of the third term to the fourth term. b Find the ratio of the nth and the (n + )th term in the sequence. arithmetic series ebookplus Digital doc SkillSHEET 5. Using elimination to solve simultaneous equations Often we have a sequence of numbers and we wish to know their sum. For an example, we return to the oil drops on the racetrack from the start of the previous section on arithmetic sequences. The distance covered by the car each second illustrated the concept of an arithmetic sequence. The total distance covered by the car is the sum of the individual dis tances covered each second. So after one second the car has travelled 0 m, after two seconds the car has 4 maths Quest Standard General mathematics for the Casio Classpad

14 travelled m = 8 m, after three seconds the car has travelled a total distance of m = 54 m, and so on. 0 metres 8 metres 6 metres 34 metres 4 metres 8 m 54 m 88 m A series, S n, is the sum of a sequence of n terms t + t + t t n. 30 m Thus: S = t S = t + t S 3 = t + t + t 3 S n = t + t + t t n - + t n - + t n. For an arithmetic sequence, the sum of the first n terms, S n, can be written in two ways:. The first term in the arithmetic sequence is a, the common differ ence is d, and the last term that is, the nth term in the sequence is l. S n = a + (a + d ) + (a + d ) + (a + (n - 3)d ) + (a + (n - )d + a + (n - )d = a + (a + d ) + (a + d ) + (l - 3d ) + (l - d ) + (l - d ) + l []. We can write the sum S n in reverse order starting with the nth term and summing back to the first term a: S n = l + (l - d ) + (l - d ) + + (a + d ) + (a + d ) + a. [] If we add equation [] and equation [] together and recognise that there are n terms each of which equal (a + l) we get: S n = (a + l) + (a + l) + n times = n(a + l) and so: S n = n (a + l ) or since l is the nth term, l = a + (n - )d, so S n = n [a + a + (n - )d ] S n = n [a + (n - )d ] The sum of the first n terms in the arithmetic sequence is given by S n = n (a + l) where a is the first term and l is the last term; or alternatively, since l = a + (n - )d, by S n = n (a + (n - )d ) where a is the first term and d is the common difference. If we know the first term, a, the common difference, d, and the number of terms, n, that we wish to add together we can calculate the sum directly without having to add up all the individual terms. It is worthwhile also to note that S n + = S n + t n +. This tells us that the next term in the series S n + is the present sum S n plus the next term in the sequence t n +. This result is useful in spreadsheets where one column gives the sequence and an adjacent column is used to give the series. Chapter 5 Sequences and series 5

15 Worked example 7 a Find the sum of the first 0 terms in the sequence t n : {, 5, 38, }. b For the sequence t n = n + 7, n {,, 3, }, find i S 5 and ii S n. ThInk WrITe ebookplus Tutorial int-0994 Worked example 7 a Write the formula for the sum of the first n terms in the arithmetic sequence. a S n = n (a + (n - )d ) Identify the variables. a =, d = 5 - = 3, n = 0 3 Substitute values of a, d and n into S 0 = 0 ( + 9 3) the formula and evaluate. S 0 = 70 b First define the sequence as follows. b On the Main screen, tap: Action Command Define Complete the entry line as: Define t(n) = n + 7 Then press E. i To find S 5, choose the Sum template by tapping: i The sum of the first 5 terms of the sequence, S 5 = 65. ) - O Complete the entry line as shown above right. Then press E. ii To find the sum from to n, use t(x) instead of t(n). ii The sum of n terms of the sequence, S n = n + 8n. remember. The sum of the first n terms of an arithmetic sequence is S n = n (a + ( n - )d) ) and so to find S n the values for a and d need to be found for the sequence whose series is required.. In general, t n + = S n + - S n. 6 maths Quest Standard General mathematics for the Casio Classpad

16 exercise 5C arithmetic series We7 Consider the following sequences and find the sums of the terms as indicated. a t n : {,, 3, }. Find S 0, S 50, S 00. b t n : {, 3, 5, }. Find S 5, S 0, S 0. c t n = 3n + 7, n {,, 3, }. Find S 5, S 0, S n. d t n = - 4n + 5, n {,, 3, }. Find S 5, S 0, S n. e t n + = t n + 5.5, t =.5. Find S 5, S 0, S 0. f t n + = t n + p, t = p. Find S 5, S 0, S 0. g The first term is 4 and the common difference is 3. Find S 4, S 6, S 64. h The first term is 4 and the common difference is - 3. Find S 4, S 9, S 4. i The first term is 50 and the 0th term is Find S 0. j The 5th term is 0 and the 8th term is 6. Find S 5, S 50, S 500. a Find the sum of the first 50 positive integers. b Find the sum of the first 00 positive integers. 3 a Find the sum of all the half-integers between 0 and 00. Note: The sequence of half-integers is {,,, 3, } b Compare your answer with that for question b. ebookplus Digital doc Spreadsheet 7 Arithmetic series 4 Find the sum of the first terms of an arithmetic sequence in which the second term is 8 and thirteenth term is 4. 5 A sequence of numbers is defined by t n : {5, 9, 3, - 3, }. a Find the sum of the first 3, 6 and 9 terms in the sequence. b Find the sum of all the terms between and including t 0 and t 5. 6 A sequence of numbers is defined by t n = n - 7, n {,, 3, }. Find a the sum of the first 0 terms b the sum of all the terms between and including t and t 40 c the average of the first 40 terms. Hint: You need to find the sum first. 7 mc Nathan decided to build up his muscles for summer by doing push-ups every day. He started with 0 push-ups a day. Every second day Nathan increased the number of pushups by. (That is, he did 0 push-ups on day and day ; push-ups on day 3 and day 4; 4 push-ups on day 5 and day 6 and so on. The total number of push-ups that Nathan did after 6 weeks is: A 68 B 70 C 840 D 60 E 4 8 mc Rachel has a box of 00 lego blocks, which she intends to use for building towers. She uses 4 blocks to build the first tower, 5 blocks to build the second tower, 6 blocks to build the third tower etc. What is the greatest number of towers that Rachel can build from the available lego blocks? A 0 B C D 3 E 4 9 Find the equation that gives the sum of the first n positive integers. 0 a Show that the sum of the first n odd integers is equal to the perfect square n. b Show that the sum of the first n even integers is equal to n + n. A sequence is 5, 7, 9,, How many consecutive terms need to be added to obtain 357? Consider the sum of the first n integers. For what value of n will the sum first exceed 000? 3 a Find the sum of all integers divisible by 3 which lie between 00 and 400. b Find the sum of all integers divisible by 6 which lie between 00 and 400. Chapter 5 Sequences and series 7

17 5d 4 The first term in an arithmetic sequence is 5 and the sum of the first 0 terms is 40. Find the common difference, d. 5 The sum of the first four terms of an arithmetic sequence is 58, and the sum of the next four terms is twice that number. Find the sum of the following four terms. 6 The sum of a series is given by S n = 4n + 3n. Use the result that t n + = S n + - S n to prove that the sequence of numbers, t n, whose series is S n = 4n + 3n is arithmetic. Find both the functional and iterative equa tions for the sequence, t n. Geometric sequences ebookplus Digital doc WorkSHEET 5. A farmer is breeding worms which he hopes to sell to local shire councils for use in the decompo sition of waste at rubbish dumps. Worms reproduce readily and the farmer expects a 0% increase per week in the mass of worms that he is farming. A 0% increase per week would mean that the mass of worms would increase by a constant factor of ( + 0 ) or.. 00 He starts off with 0 kg of worms. By the beginning of the second week he will expect 0. = kg of worms, by the start of the third week he would expect. = 0 (.) =. kg of worms, and so on. This is an example of a geometric sequence. A geometric sequence is the sequence where each term is obtained by multiplying the preceding term by a certain constant factor. The first term is 0 and the common factor here is.0, which rep resents a 0% increase on the previous term. We can put the results of the above example into a table: n t n t n 0 (.) (.) 3 0 (.). 4 0 (.) n 0 (.) n - 0 (.) n - From this table we can see that t =. t, t 3 =. t and so on. In general: t n + =. t n The common factor or common ratio whose value is. for this example can be found by dividing any two successive terms: t n+. t n A geometric sequence, t, can be written in terms of the first term, a, and the common ratio, r. Thus: t : {a, ar, ar, ar 3,, ar n -, } The first term t = a, the second term t = ar, the third term t 3 = ar and consequently the nth term, t n is ar n -. 8 maths Quest Standard General mathematics for the Casio Classpad

18 Worked Example 8 For a geometric sequence: t n = ar n - where a is the first term and r the common ratio, given by n + r = t tn If we consider three consecutive terms in a geometric sequence, x, y and z, then y z = r = x y where r is the common factor. Thus the middle term, y, called the geometric mean, can be calculated in terms of the outer two terms, x and z. For a geometric sequence, x, y, z, : y = xz State whether the sequence is geometric by finding the ratio of successive terms: t: {, 6, 8, }. If it is geometric, find the next term in the sequence, t 4, and the nth term for the sequence, t n. Think Find the ratio t. t Find the ratio t t 3. t t t t 3 Write = 6 = 3 = 8 6 = 3 3 Compare the ratios and make your conclusion. Since t t 4 Since the sequence is geometric, to find the fourth term, multiply the preceding (third) term by the common ratio. 5 Write the general formula for the nth term of the geometric sequence. 6 Identify the values of a and r. a = ; r = 3 = t 3 t 7 Substitute the values of a and r into the general formula. t n = 3 n - = 3, the sequence is geometric with the common ratio r = 3. t 4 = t 3 r = 8 3 = 54 t n = ar n - Worked Example 9 Find the nth term and the 0th term in the geometric sequence, where the first term is 3 and the third term is. Think Write the general formula for the nth term in the geometric sequence. State the value of a (the first term in the sequence) and the value of the third term. Write t n = ar n - a = 3; t 3 = Chapter 5 Sequences and series 9

19 3 Substitute all known values into the general formula. = 3 r 3-4 Solve for r (note that there are possible solutions). r = 3 = 4 5 Substitute the values of a and r into the general equation. Since there are possible values for r, you must show both expressions for the nth term of the sequence. 6 Find the 0th term by substituting n = 0 into each of the two expressions for the nth term. = 3 r r = ± 4 = ± So t n = 3 n -, or t n = 3 ( - ) n - When n = 0, t 0 = (using r = ) = 3 9 = 536 or t 0 = 3 ( - ) 0 - (using r = - ) = 3 ( - ) 9 = Worked example 0 The fifth term in a geometric sequence is 4 and the seventh term is Find the common ratio, r, the first term, a, and the nth term for the sequence. ThInk Method : Technology-free Write the general rule for the nth term of the geometric sequence. Use the information about the 5th term to form an equation. Label it []. 3 Similarly, use information about the 7th term to form an equation. Label it []. 4 Solve equations simultaneously: Divide equation [] by equation [] to eliminate a. WrITe t n = ar n - 5 Solve for r. r = Since there are two solutions, we have to perform two sets of computations. Consider the positive value of r first. Substitute the value of r into either of the two equations, say equation [], and solve for a. 7 Substitute the values of r and a into the general equation to find the expression for the nth term. When n = 5, t n = 4 4 = a r 5-4 = a r 4 [] When n = 7, t n = = a r = a r 6 [] [ ] ar6 056 gives =. [] ar 4 4 r = ± 004. = ± 0. If r = 0. Substitute r into []: a ( 0. ) 4 = a = 4 8 Now consider the negative value of r. If r = - 0. a = = 8750 The nth term is: t n = 8750 (0.) n - ebookplus Tutorial int-0877 Worked example 0 30 maths Quest Standard General mathematics for the Casio Classpad

20 9 Substitute the value of r into either of the two equations, say equation [], and solve for a. (Note that the value of a is the same for both values of r.) 0 Substitute the values of r and a into the general formula to find the second expression for the nth term of the sequence. Method : Technology-enabled Repeat steps 3 from Method to obtain equations [] and []. Substitute r into [] a ( 0. ) 4 = a = 4 a = = 8750 The nth term is: t n = 8750 ( - 0.) n - or t n = 8750 (0.) n - 4 = a r 4 [] 056. = a r 6 [ ] Solve equations [] and [] simultaneously. On the Main screen, set up the two equations by using the soft keyboard and tapping: ) {N Complete the entry lines as shown at right and press E. 3 Write the solution. Solving 4 = a r and 0.56 = a r 6 for a and r gives a = 8750 and r = - 0., or a = 8750 and r = 0. 4 Substitute the values of r and a into the general formula to find the expression of the nth term of the sequence. Note: Since there are two possible values of r, there will be two different expressions for t n. The nth term is: t n = 8750 ( 0.) n -, or t n = 8750 (0.) n - REMEMBER. A geometric sequence is one where each successive term is obtained by multiplying the preceding term by the constant number. This number is called the common ratio and is given the symbol r. Thus t n + = r for all values of n. The first term in the t sequence is given the symbol a. n. If x, y, z are successive terms in the geometric sequence then y is called a geometric mean and is given by y = xz. 3. A geometric sequence can be written as a, ar, ar, and so the nth term t n is: t n = ar n - using the function notation, or t n + = rt n, t = a using the iterative notation. Chapter 5 Sequences and series 3

21 exercise 5d Geometric sequences We8 State which of the following are geometric sequences by finding the ratio of successive terms. For those which are geometric, find the next term in the sequence, t 4 and the nth term for the sequence, t n. a t n : {3, 6, 9, } b t n : {4,, 36, } c t n : {3, 6,, } d t n : {4, 6, 9, } e t n : { - 3,, 3 f t n: {, - 6, 8, } g t n : {, h t n: { 3, 3, 3, } 4 i t n: { 3, 3, 9, } 4 4 j t n : { 4, - 3, 9, } k t n: {p, 4p, 8p 3, } For each of the following: i show that the sequence is geometric ii find the nth term and consequently the 6th and the 0th terms. a t : {5, 0, 0, } b t : {, 5,.5, } c t : {, - 3, 9, } d t : {, - 4, 8, } e t : {.3, 3.45, 5.75, } f t : {,,, } g t : {,,, } h t : 3 48 {3, -,, } i t : {x, x4, 9x 7, } j t : {,, 4, } 3 x x x 3 We9 Find the nth term and the 0th term in the geometric sequence where: a the first term is and the third term is 8 (Why are there two possible answers?) b the first term is and the third term is 4 (Why are there two poss ible answers?) c the first term is 5 and the fourth term is 40 d the first term is - and the second term is e the first term is 9 and the third term is. (Why are there two possible answers?) 8 4 Find the 4th term in the geometric sequence where the first term is 6 and the 7th term is mc A geometric sequence is such that a > 0 and r < 0. Which of the graphs below could represent the first seven terms of this sequence? A t n B t n C t n n n ebookplus Digital doc Spreadsheet 7 Geometric sequences n D t n E n t n n 3 maths Quest Standard General mathematics for the Casio Classpad

22 6 A crystal measured.0 cm in length at the beginning of a chemistry experiment. Each day it increased in length by 3%. The length of the crystal after 4 days growth is closest to: A.4 cm B 6.7 cm C 7.0 cm D 7.6 cm E 8. cm [ VCAA 006] Exam tip This question required students to determine the length of a crystal after 4 days growth. Only 5% of students gave the correct response. Thirty-nine per cent of students chose wrong option that corresponded to the length of the crystal at the start of the 4th day, which only allowed for 3 days growth. [Assessment report 006] 5E 7 Find the nth term in the geometric sequence where the first term is 3 and the fourth term is 6. 8 For the geometric sequence 3, m, n, 9,, find the values for m and n. 9 Consider the geometric sequence t : {6, m, 8, n, }. Find the values of m and n, if it is known that both are positive numbers. 0 For the geometric sequence a, 5, b, ,, find the values of a and b, given that they are positive numbers. WE0 The third term in a geometric sequence is 00 and the fifth term is 400. Find the common ratio, r, the first term, a, and the nth term for the sequence. If t = and t 5 = 7, find the first term, a, the common factor, r, and hence the nth term for the 6 geometric sequence. 3 Find the value of x such that the following sequence forms a geometric progression: x -, 3x + 4, 6x Insert three terms in between 8, _, _, _, such that the sequence of numbers is geometric. 3 5 The difference between the first term and the second term in a geometric sequence is 6. The difference between the second term and the third term is 3. a Calculate the difference between the third term and the fourth term. b Find the nth term in the sequence. 6 The first two terms in a geometric sequence are 0, 4, and the kth term is Find the value for k. Geometric series When we add up or sum the terms in a sequence we get the series for that sequence. If we look at the geometric sequence {, 6, 8, 54, } where the first term t = a = and the common ratio is 3 we can quickly calculate the first few terms in the series of this sequence. S = t = S = t + t = + 6 = 8 S 3 = t + t + t 3 = = 6 S 4 = t + t + t 3 + t 4 = = 80 In general the sum of the first n terms is: S n = t + t + t t n - + t n - + t n. For a geometric sequence the first term is a, the second term is ar, the third term is ar and so on up to the nth term which is ar n. Thus: S n = a + ar + ar + + ar n ar n + ar n [] If we multiply equation [] by r we get: rs n = ar + ar + ar 3 + ar n - + ar n - + ar n [] Chapter 5 Sequences and series 33

23 Note that on the right-hand side of equations [] and [] all but two terms are common, namely the first term in equation [], a, and the last term in equation [], ar n. If we take the difference between equa tion [] and equation [] we get: rs n - S n = ar n - a [] - [] (r - )S n = a(r n - ) ar ( n ) Sn = ; r (r cannot equal ) r We now have an equation which allows us to calculate the sum of the first n terms of a geometric sequence. The sum of the first n terms of a geometric sequence is given by: ar ( n ) Sn = r ; r where a is the first term of the sequence and r is the common ratio. Worked Example Find the sum of the first 5 terms (S 5 ) of these geometric sequences. a t n : {, 4, 6,...} b t n = () n -, n {,, 3,...} c t n + = t 4 n, t = Think Write a Write the general formula for the sum ar ( n ) a S n = of the first n terms of the geometric r sequence. Write the sequence. t n : {, 4, 6....} 3 Identify the variables: a is the first term; r can be established by finding the ratio; n is known from the question. 4 Substitute the values of a, r and n into the formula and evaluate. a = ; r = 4 = 4; n = 5 S 5 = 4 ( 5 ) 4 = 04 3 = 34 b Write the sequence. b t n = () n, n {,, 3,...} 3 Define the sequence as follows. On the Main screen, tap: Action Command Define Complete the entry line as: Define t(n) = n - Then press E. To find S 5, choose the Sum template by tapping: ) - O Complete the entry line as shown at right. Then press E. Write the solution. The sum of the first 5 terms of the sequence is S 5 = Maths Quest Standard General Mathematics for the Casio ClassPad

24 c Write the sequence. c t n + = 4 t n, t = This is an iterative formula, so the coefficient of t n is our r; a = t ; n is known from the question. 3 Substitute values of a, r and n into the general formula for the sum and evaluate. r = 4 ; a = ; n = 5 S 5 = = [( ) 5 ] 4 4 = 34 5 ( ) The infinite sum of a geometric sequence where r < When the constant ratio, r, is less than but greater than -, that is, {r: - < r < }, each successive term in the sequence gets closer to zero. This can readily be shown with the following two examples. g: {, -,, 4, } where a = and r = h: {40,,, } where a = 40 and r = In both the examples, successive terms approach zero as n increases. In the second case the approach is more rapid than in the first and the first sequence alternates positive and negative. A simple investigation with a spreadsheet will quickly reveal that for geometric sequences with the size or magnitude of r < the series eventually settles down to a near constant value. We say that the series converges to a value S which is the sum to infinity of all terms in the geometric sequence. We can find the value S by recognising that as n the term r n 0, provided r is between - and. We write this technically as - < r < or r <. The symbol r means the magnitude or size of r. Using our equation for the sum of the first n terms: S n ar ( n ) = r ; r Taking - as a common factor from the numerator and denominator: S n a( rn) = r As n, r n 0 and hence - r n. Thus the top line or numer ator will equal a when n : a S = ; r < r We now have an equation which allows us to calculate the sum to infinity, S of a geometric sequence. The sum to infinity S of the geometric sequence is given by: S a = ; r r < where a is the first term of the sequence and r is the common ratio whose magnitude is less than one. Chapter 5 Sequences and series 35

25 Worked example a Find the sum to infinity for the sequence t n : {0,, 0.,...}. b Find the fourth term in the geometric sequence whose first term is 6 and whose sum to infinity is 0. ThInk WrITe a Write the formula for the nth term of the a t n = ar n - geometric sequence. From the question we know that the first a = 0, r = 0. term, a, is 0 and r = Write the formula for the sum to infinity. a S ; r < r 4 Substitute a = 0 and r = 0. into the formula and evaluate. S S 0 = = = = b Write the formula for the sum to infinity. b a S ; r < r From the question it is known that the infinite sum is equal to 0 and that the first term a is 6. Write down this information. a = 6; S = Substitute known values into the formula. 0 = r 4 Solve for r. 0( - r) = 6 0-0r = 6 0r = 4 r = Write the general formula for the nth term of the geometric sequence. 6 To find the 4th term substitute a = 6, n = 4 and r = 0.4 into the formula and evaluate. t n = ar n - t 4 = 6 (0.4) 3 = remember. The sum of the first n terms in a geometric sequence is: a Sn = ( r n ) with r r a or Sn = (rr n ) with r r. When the magnitude of r is less than one, that is, - < r <, the sum of a geometric sequence to infinity, S is given by: a S = r 36 maths Quest Standard General mathematics for the Casio Classpad

26 exercise 5e Geometric series We Consider the following sequences and find the terms indicated. a t n : {,, 4, }. Find S 5, S 0, S 0. b t n : {, 3, 9, }. Find S 5, S 0, S 0. c t n = 3( - ) n -, n {,, 3, }. Find S 5, S 0, S 0. d t n = - 4(.) n -, n {,, 3, }. Find S, S 0, S 0. e t n + = t n, t = 3. Find S, S 5, S 0. f t n + = t n, t = 3. Find S, S 5, S 0. g The first term is 3000 and the common ratio is.05. Find S 4, S 6, S 64. h The first term is 400 and the common ratio is -.. Find S 4, S 9, S 4. i The first term is 0; every other term is obtained by multiplying the preceding term by 5. Find S 5, S 0. j The first term is - ; every other term is obtained by multiplying the preceding term by. Find S 5, S 0. Consider the following geometric sequences and find the terms indicated. a The first term is 440 and the th term is 880. Find S 6. b The 5th term is and the 8th term is 8. Find S, S 0, S 0. 3 Find the sum of the first terms of a geometric sequence in which the second term is 8 and the fifth term is What minimum number of terms of the series must be taken to give a sum in excess of 00? 5 The sum of the first four terms of a geometric sequence is 3, and the sum of the next four terms is 65 times that number. Find the sum of the following four terms. 6 Find the sum of all powers of between 500 and Find the sum of all powers of 4 between 500 and mc In preparation for an upcoming school production Lena spent hours reading and memorising the script on the first day after receiving it. The next day she spent another hour and 48 minutes on the same task and the day after that she spent hour and 37. minutes. If the time that Lena spent daily working on memorising the script followed the same pattern for two weeks, the total time that she thus spent was closest to: A 4 hours 5 minutes B 4 hours 5 minutes C 5 hours 5 minutes D 6 hours E 6 hours 5 minutes 9 We a Find the sum to infinity for the following geometric sequences. a t n : {,, 4, } b t n: {,, 4, 8, } c t n : {, 3, 9, } e t n : {, 3, 4 9, 8 9, } d t n: {, 3, 4 9, } ebookplus Digital doc Spreadsheet 7 Geometric series 0 For the infinite geometric sequence {,,, }, find the sum to infinity. Consequently, find 4 8 what proportion each of the first three terms contributes to this sum as a percentage. For the infinite geometric sequence {,,, }, find the sum to infinity. Consequently, find 4 6 what proportion each of the first three terms contributes to this sum as a percentage. For the infinite geometric sequence {, 3, 9, }, find the sum to infinity. Consequently, find 4 6 what proportion each of the first three terms contributes to this sum as a percentage. Chapter 5 Sequences and series 37

27 n 3 A sequence of numbers is defined by t n = 3, n {,, 3, }. a Find the sum of the first 0 terms. b Find the sum of all the terms between and including t and t 40. c Find the sum to infinity, S. 4 A sequence of numbers is defined by t n : {9, - 3,, }. a Find the sum of the first 9 terms. b Find the sum of all the terms between and including t 0 and t 5. c Find the sum to infinity, S. 5 The first term of the geometric sequence is 5 and the fourth term is Find the sum to infinity. 6 The sum of the first four terms of a geometric sequence is 30 and the sum to infinity is 3. Find the first three terms of the sequence. 7 For the geometric sequence 5+ 3, 5 3,, find the common factor, ebookplus r, and the sum of the infinite series, S. 8 If + 3x + 9x + =, find the value of x. 3 9 We b The first term in a geometric sequence is 4 and S = 6. Find the common factor, r. 0 If the common ratio for a geometric sequence is 0.99 and the sum to infinity is 00, what is the value of the first and second terms in the sequence? A student stands at one side of a road 0 metres wide, and walks half-way across. The student then walks half of the remaining distance across the road, then half the remaining distance again and so on. a Will the student ever make it past the other side of the road? b Does the width of the road affect your answer? Digital doc WorkSHEET 5. 5F applications of sequences and series ebookplus Interactivity int-0806 Applications of sequences and series This section consists of a mixture of problems where the work covered in the first five exercises is applied to a variety of situations. The following general guidelines can assist you in solving the prob lems.. Read the question carefully.. Decide whether the information suggests an arithmetic or geometric sequence. Check to see if there is a constant difference between suc cessive terms or a constant ratio. If there is neither, look for a simple number pattern such as the difference between successive terms changing in a regular way. 3. Write the information from the problem using appropriate notation. For example, if you are told that the 5th term is, write t 5 =. If the sequence is arithmetic, you then have an equation to work with, namely: a + 4d =. If you know the sequence is geometric, then ar 4 =. 4. Define what you have to calculate and write an appropriate formula or formulas. For example, if you have to find the 0th number in a sequence which you know is geometric, you have an equation: t 0 = ar 9. This can be calculated if a and r are known or can be established. 5. Use algebra to find what is required in the problem. 38 maths Quest Standard General mathematics for the Casio Classpad

28 Worked example 3 In 970 the cost of megabyte of computer memory was $05. In 980 the cost for the same amount of memory had reduced to $45 and by 990 the cost had dropped to $. a What was the cost of megabyte of memory in the year 000? ebookplus Tutorial int-0878 Worked example 3 b How much memory, in megabytes, could you buy for $0 in the year 00 based on the current trend? ThInk a Present the given information in a table. a WrITe Year Cost ($) 05 45?? Study the table. The information suggests a geometric sequence for the cost at each ten-year interval. Verify this by checking for a constant ratio between successive terms. 3 To find the cost in the year 000, find the fourth term in the sequence by multiplying the preceding (third) term by the common ratio. 4 Interpret the result and clearly answer the question. b If the cost of megabyte can be found in the year 00 then the amount of memory purchased for $0 can be determined. To find the predicted cost in the year 00 the fifth term in the sequence needs to be determined. Take the reciprocal of t 5 to get the amount of memory per dollar. 3 Find the amount of memory that can be purchased for $0. b = and 45 = so the three terms form a geometric sequence with common ratio r =. 45 t 4 = t 3 r t 4 = 45 = 45 = 0.0 In the year 000 one would have paid about cents for a megabyte of memory. t 5 = t 4 r = = of a dollar per megabyte 05 The amount of memory per dollar is 05 megabytes. So $0 would buy 0 05 = 0 50 megabytes. Worked example 4 Express the recurring decimal as a proper fraction. ThInk WrITe Express the given number as a geometric series = State the values of a and r. a = 0.3 and r = = Find the sum to infinity, S : Write the formula for the sum to infinity. a S = r 03. Chapter 5 Sequences and series 39

29 4 Substitute values of a and r into the formula and simplify. 5 Multiply both numerator and denominator by 00 to get rid of the decimal point. 03. S = 0. 0 S = S = 3 99 REMEMBER To solve problems, use the following guidelines.. Identify the type of the sequence by checking whether there is a common difference, or a common ratio.. Translate given information into mathematical statements, using appropriate notation. 3. Define what you have to find and write appropriate formula(s). 4. Use algebra to find what is required. Exercise 5F Applications of sequences and series WE 3 In 970 the Smith family purchased a small house for $ Over the following years, the value of their property rose steadily. In 975 the value of the house was $ and in 980 it reached $ a Assuming that the pattern continues through the years, find (to the nearest dollar) the value of the Smiths house in i 985, ii 995. b By what factor will the value of the house have increased by the year 00, compared to the original value? An accountant has been working with the same company for 5 years. She commenced on a salary of $8 000 dollars and has received a $500 increase each year. a What type of sequence of numbers does her annual income follow? b How much did she earn in her 5th year of employment? c How much has she earned from the company altogether? d What was her percentage increase at the end of i her first and ii her fourteenth year of employment? 3 A chemist has been working with the same company for 5 years. He commenced on a salary of $8 000 dollars and has received a 4% increase each year. a What type of sequence of numbers does his annual income follow? b How much did he earn in his 5th year of employment? c How much has he earned from the company altogether? d What was his increase in salary at the end of i his first and ii his fourteenth year of employment? 4 A biologist is growing a tissue culture in a Petri dish. The initial mass of the culture was 0 milligrams. By the end of the first day the culture was a mass of 8 milligrams. 40 Maths Quest Standard General Mathematics for the Casio ClassPad

30 a Assuming that the daily growth is arithmetic, find the mass of the culture after the second, third, tenth and nth day. b On what day will the culture mass first exceed 00 milligrams? c Assuming that the daily growth is geometric, find the mass of the culture after the second, third, tenth and nth day. d On what day will the culture mass first exceed 00 milligrams? 5 Logs of wood can be stacked so that there is one more log on each descending layer than on the previous layer. The top row has 6 logs and there are 0 rows. a How many logs are in the stack altogether? b The logs are to be separated into two equal piles. They are sepa rated by removing logs from the top of the pile. How many rows down will workers take away before they remove half the stack? 6 As I was going to St Ives I met a man with seven wives. Every wife had seven sacks, Every sack had seven cats, Every cat had seven kits. Kits, cats, sacks and wives, How many were coming from St Ives? Note: This is a variation on the original riddle, which asks How many were going to St Ives. 7 Thoughtful Frank has 00 movie tickets to give away to people at a local shopping centre. He gives the first person one ticket, the next person two tickets, the third person three tickets and so on until he can no longer give the nth person n tickets. How many tickets did the last lucky person receive? How many tickets did Frank have left? 8 Kind-hearted Kate has 00 movie tickets to give away to people at the shopping centre. She gives the first person one ticket, the next person two tickets, the third person four tickets and so on following a geometric progression until she can no longer give the nth person (n - ) tickets. How many tickets did the last lucky person receive? How many tickets did Kate have left? 9 The King of Persia, so the story goes, offered Xanadu any reward to secure the safety of his kingdom. As his reward, Xanadu requested a chessboard with one grain of rice on the first square, two grains on the second, four on the third and so on until the 64th square had its share of rice deposited. a Find the total number of grains of rice that the king needed to supply. b If each grain of rice weighs 0.0 grams, how many kilograms of rice does this represent? (Note: There are 0 3 grams in kilo gram.) 0 As legend has it, the King of Constantinople offered Xanadu s cousin Yittrius any reward to secure the safety of his city. This Yittrius accepted: she requested a chessboard with one grain of rice on the first square, three grains of rice on the second square, five grains of rice on the third square and so on until the 64th square had its share of rice deposited. a Find the total number of grains of rice that the king needed to supply. b If each grain of rice weighs 0.0 gram, how many kilograms of rice does this represent? (Note: There are 0 3 grams in kilo gram.) A student is 3.0 m from the door to a classroom and decides that he will take a.0 m step followed by a step of half that distance, and half again and so on until he gets to the classroom door. Show that he will never get any closer than one metre from the door. A hiker walks 36 km on the first day and that distance on the second. Every day thereafter 3 she walks of the distance she walked on the day before. Will the hiker cover the distance of 3 00 km to complete the walk and on what day will she complete the task? Chapter 5 Sequences and series 4

31 3 We 4 Recurring decimals can be expressed as rational numbers. Find the fraction equivalent of the following recurring decimal numbers by writing the decimal number as a sum of infinite terms. a 0. = b c d e f. g In 990, 00 students enrolled for a hypocorisma subject at a local university. Each subsequent year for the next decade the enrolment increased by 0%. a Find the number of students enrolled in hypocorisma in 995. b Over the course of the decade find the total number of students who had enrolled in hypocorisma. 5 For tax purposes, the value of a computer used for a business depreciates by 8.5% of the initial cost each year. For economic reasons the business sells its computers when they first depreciate to less than half their initial value. After how many years will a computer used by this business be sold? 6 The side lengths of a right-angled triangle form the successive terms of an arithmetic sequence. The perimeter of the triangle is 7 m. What are the side lengths of the triangle? 7 A circular board is divided into a series of concentric circles of radius cm, cm, 3 cm and 4 cm as shown at right. a Find the areas of each of the successive shaded regions and show that they form an arithmetic 4 cm progression. b A dart is fired at the board at random and hits 3 cm the board. What is the probability of striking each of the four regions of the board? cm (Note: The probability of striking a region = area of cm region total area.) 8 A bullet is fired vertically up into the air. In the first second it has an average speed of 80 m/s; that is, it travels 80 m up into the air during the first second. Each second its speed diminishes by m/s. Thus during the nd second the bullet has an average speed only 68 m/s and accordingly travels 68 m further up into the air. a Find an equation for the average speed of the bullet for the nth second that it is in the air. b Find the time when the average speed of the bullet is equal to zero. c Find the maximum height of the bullet above where it was fired. 9 Coffee cools according to Newton s Law of Cooling in which the temperature of the coffee above room temperature drops by a con stant fraction each unit of time. The table below shows the tem perature of a cup of coffee in a room at 0 C each minute after it was made. Remember to sub tract the room temperature from the temperature of the coffee before you do your calcu lations. Time (min) Temperature ( C) maths Quest Standard General mathematics for the Casio Classpad

32 The person who made the coffee will drink it only if it has a temperature in excess of 50 C. What is the minimum time after the cup of coffee has been made before it will not be drinkable? 0 Two arithmetic sequences, t n and u n, are multiplied together. That is, each term is multiplied by the other to form a new term. t n = n - 3, n {,, 3, } and u n = 3n, n (,, 3, } Show that the new sequence of numbers t u, t u, t 3 u 3, is an arithmetic series and hence find the arithmetic sequence for that new series. (Hint: For a sequence, a n, with a series A n, a n = A n - A n -.) Chapter 5 Sequences and series 43

33 SummarY Describing sequences A sequence is a string of numbers or expressions. It may contain a finite or infinite number of terms and may or may not follow a recognisable pattern. A sequence can be described in a number of ways.. As a list t n : {, 7,, 6, } (note that t 3 = ). As a function: t n = n - n, n {,, 3, } (note that t 5 = 5-5 = - 5) 3. As a recursive or iterative formula: t n + = t n - 3, t = 6 (note that t = 6-3 = 9) The logistic equation is a model of population growth of the general form: t n + = at n ( t n ), where 0 < t 0 < and a is a constant. A convergent sequence is a sequence whose terms settle at a certain fixed number, x, called the limit of the sequence. This can be written as t n x. A sequence whose terms grow further and further apart is called divergent. That is, a sequence is divergent if t n, or t n - as n. A sequence whose terms fluctuate between two (or more) values is called oscillating. Arithmetic sequences An arithmetic sequence is one whose successive terms have a common difference. This common difference is given the symbol d. Thus t n + - t n = d for all values of n. The first term in the sequence is given the symbol a. If x, y, z are successive terms in an arithmetic sequence then the middle term (y) is called an arithmetic mean and is equal to the average of the two outer terms (x and z): x z y = + An arithmetic sequence can be written as a, a + d, a + d, and so the nth term, t n, is: t n = a + (n - )d using the function notation, or t n + = t n + d, t = a using the iterative notation. Arithmetic series The sum of the first n terms of the arithmetic sequence is given by In general, t n + = S n + - S n. Geometric sequences n Sn = a + n d ( ( ) ) A geometric sequence is one in which each successive term is obtained by multiplying the preceding term by a constant number. This number is called the common ratio and is given the symbol r. Thus t n+ = r for all values of n. The first term in the sequence is given the symbol a. tn If x, y, z are successive terms in an arithmetic sequence then y is called a geometric mean and is given by y = xz. A geometric sequence can be written as a, ar, ar, and so the nth term, t n, is t n = ar n - using the function notation, or t n + = rt n, t = a using the iterative notation. 44 maths Quest Standard General mathematics for the Casio Classpad

34 Geometric series The sum of the first n terms in a geometric sequence is given by a( rn) Sn = with r r ar ( n ) or Sn = with r r When the magnitude of r is less than one, that is, - < r <, the sum of a geometric sequence a to infinity S is given by S =. r Applications of arithmetic and geometric sequences and series To solve problems, use the following guidelines.. Identify the type of sequence by checking whether there is a common difference, or a common ratio.. Translate given information into mathematical statements, using appropriate notation. 3. Define what you have to find and write appropriate formula(s). 4. Use algebra to find what is required. Chapter 5 Sequences and series 45

35 chapter review Multiple choice Consider the sequence t n + = t n + 4; t 3 =. The second term in the sequence is: A 0 B 6 C 8 D 4 E 8 A series is listed as 3, 0,, 36, The next term in the series is: A 5 B 5 C 53 D 54 E 55 3 Paula started a stamp collection. She decided to buy a number of new stamps every week. The number of stamps bought in the nth week, t n, is defined by the equation t n = t n + t n, where t = and t =. The total number of stamps in her collection after five weeks is: A 8 B C 5 D 9 E 4 [ VCAA 006] Exam tip This question was an application of a Fibonacci-related sequence, which required students to determine and then sum the first five terms in the sequence. Option A, which was chosen by 9% of students, corresponded to the value of the fifth term in the sequence. [Assessment report 006] 4 The 3rd term in the sequence of numbers {7, 3, -, } is: A - 88 B - 8 C - 74 D - 83 E 90 5 Consider the arithmetic sequence 5, a, 4, b. The numerical value of the expression a - 3b is: A - 60 B - 64 C - 67 D - 7 E 7 6 Consider the arithmetic sequence x - y, 3x - 4y, 4x - 7y, An expression for y in terms of x is: A y = x B y = - x C y = - x D y = x E y = - 3x 7 A car is accelerating such that in the st second it travels.0 metres, in the nd second it travels 3.5 metres, in the 3rd second it travels 5.0 metres, and so on for a total of 5 seconds. The total distance travelled by the car is: A 630 m B m C 87.5 m D 375 m E 35 m 8 The sum of the first four terms in an arithmetic sequence is 70. The sum of the first six terms is 63. The sixth term of the sequence is equal to: A - 4 B - 7 C 0 D 7 E 4 9 For a geometric sequence, the 4th term is 5 and the 7th term is The second term in the sequence is: A -.5 B -.5 C 0.5 D E t n The graph above shows the first six terms of a sequence. This sequence is best described as: A geometric with - < r < 0 B geometric with 0 < r < C geometric with r > D arithmetic with d < 0 E arithmetic with 0 < d < [ VCAA 005] The sum of an infinite geometric sequence is 5.6 with the common ratio equal to 0.0. The sum of the first four terms of the geometric sequence is closest to: A 5.0 B 5. C 5.4 D 5.6 E 5.8 The sum of the first 0 terms of a geometric sequence is 400. The next term in the sequence is 3 times the previous term. The first term in the sequence is: A 7 73 B D E n C A healthy eating and gym program is designed to help football recruits build body weight over an extended period of time. Roh, a new recruit who initially weighs 73.4 kg, decides to follow the program. In the first week he gains 400 g in body weight. In the second week he gains 380 g in body weight. In the third week he gains 36 g in body weight. If Roh continues to follow this program indefinitely, and this pattern of weight gain 46 Maths Quest Standard General Mathematics for the Casio ClassPad

36 remains the same, his eventual body weight will be closest to: A 74.5 kg B 77. kg C 77.3 kg D 80.0 kg E 8.4 kg [ VCAA 006] 4 The sequence t n = 8 3 n - and the sequence s n = - 0.n are combined to form the ratio t n. s When n = 9 the value of the ratio is: n A D Exam tip This question required students to model an athlete s weight gain over an extended period of time using an infinite geometric sequence. Only 33% of students gave the correct response. The relatively even distribution responses across the incorrect options suggests that the majority of students guessed their answer to this question. [Assessment report 006] 7 B 70 C 43 8 E 0 8 Short answer Write the iterative definition for each of the following sequences: a {7,, 9, 35, 67, } b { -, 5, 6, 677, } For each of the following values of a and t 0 : a a = 0.3, t 0 = 0.4 b a =.8, t 0 = 0.6 c a = 3., t 0 = 0.8 i use the logistic equation t n + = at n ( t n ), to generate a sequence of 5 terms ii state whether the sequence is convergent, divergent, or oscillating (you may need to generate more terms first) iii if the sequence is convergent, state its limit. 3 For the arithmetic sequence where t 3 = 0 and t 6 = 478, find a the functional rule for the nth term in the sequence b the iterative rule for the sequence. 4 A car at a racetrack starts from rest and travels 0.5 m in the st second and.0 m in the nd second following an arithmetic progression in the distances covered each subsequent second. a How far will it travel during the 0th second? b After 0 seconds of motion, how far will it have travelled in total? c To the nearest whole second, how long will it take to travel 000 m ( km)? 5 Noel is an enthusiastic runner. He decides to follow a training program over several weeks. The distance (in kilometres) that Noel runs each day forms an arithmetic sequence. The distances for the first three days of this training program are shown below. Day 3 Distance (km) a What distance (in kilometres) will Noel run on Day 5? b On which day will Noel first run more than 7 km? c Determine the total distance that Noel runs from Day 3 to Day inclusive. Write your answer in kilometres, correct to decimal place. Exam tip Some students listed, then added, all the terms from t 3 to t, and received full marks. Others stopped at t and did not gain full marks. A number of students understood that S S was required, but many incorrectly said this was equal to S 9 or S 0. [Assessment report 005] d An expression for the nth term of this sequence can be written as t n = 0.3n + b Determine the value of b. [ VCAA 005] Exam tip A common incorrect answer was 4. (i.e. the value of a). [Assessment report 005] 6 Consider the geometric sequence g n =.4.5 n - ; n =,, 3, a Find the first and 5th terms of the sequence. b Find the sum of the first 0 terms. c Find the percentage contribution that the 0th term makes to this sum. 7 At Bugas Heights a radiation leak in a waste disposal tank potentially exposes staff to a 000 milli-rem h dose on the first day of the accident, a 800 milli-rem h dose on the second day after the accident and a 640 milli-rem h dose on the third day following the accident. a Assuming a geometric sequence, find the amount of potential exposure dose by the 0th day. b Find the total potential exposure dose in the first 5 days. 8 Australian Heating is a company that produces heating systems. The number of heating systems produced annually is modelled by an increasing geometric sequence. The number of heating systems produced in each of the first three years is shown in the table below. Year 3 Number of heating systems produced Chapter 5 Sequences and series 47

37 a Show that the common ratio, r, of this geometric sequence is.. Exam tip Most students were able to show some appropriate division or multiplication that showed the common ratio was.. [Assessment report 004] b What is the annual percentage increase in the number of heating systems produced each year? Exam tip Many students got this wrong. A common incorrect answer was 0%. [Assessment report 004] c How many heating systems will be produced in year 5? Write your answer correct to the nearest whole number. Exam tip Some students did not round this answer to a whole number of heating systems. [Assessment report 004] d The number of heating systems produced annually continues to follow this pattern. In total, how many heating systems will they produce in the first 0 years of operation? [ VCAA 004] Exam tip A number of students did not round this answer to a whole number of heating systems. Several tabulated the sequence and added the terms rather than use the appropriate series formula. Some students found only the number of heating systems produced in the tenth year. [Assessment report 004] 9 The infinite sum of a geometric sequence is 99 and the first term is 0. Find the common ratio for the sequence. 0 Find the sum of the following expressions: a b a For the sequence defined by t n + = (t n + ); t n t =, find the values of t, t 3, t 0, using a spreadsheet or calculator. b The terms in this sequence give successive approximations of a real number. What is this number? Find the fraction equivalent of the following recurring decimals: a 0. b Extended Response Consider a square of side length units. a What is the perimeter of the square? b Each of the four midpoints form the vertices of a new square inscribed within the original square. Find the perimeter of this new square. c Repeat the process to find the perimeter of a third square inscribed within the second. d Give an expression for the perimeter of the nth square. Consider the following iterative definitions: a t n + = t n 3 4, t = 8 b t n + = at n, t = b c t n + = 3t n.5, t = 0.5 If each of these definitions is used to generate a sequence of numbers: i decide whether the sequence is arithmetic, geometric or neither, and ii find its fourth term. 3 In January 004, Rachel and Nathan inherited a small trout farm from their Uncle Michael. They were told that in any given year the trout population, p n +, could be easily calculated using the formula p n + = 0.5p n ( p n ), where p n is the number of trout (in thousands) in the preceding year. They were also told that on the day of their inheritance the farm housed 800 fish. a Use the above formula to predict (to the nearest whole number) the size of the fish population on Rachel and Nathan s farm for the next three years; that is, for i January 005 ii January 006 iii January Maths Quest Standard General Mathematics for the Casio ClassPad

38 b What will happen to the size of the fish population if it continues to change according to the formula? How long will it take? After extensive research, Rachel and Nathan decided to modernise their newly acquired farm. A new feeding system and other improvements were installed, and were completed within the first two years (that is, by January 006). As a result, in any given year the trout population, p n +, could now be calculated using the formula p n + =.6p n ( p n ), where p n is the number of trout (in thousands) in the preceding year. c Use this new formula, and the figure obtained in part a ii (that is, the trout population in January 006) as your starting point to predict the size of the trout population for January 007 and January 008. d Do Rachel and Nathan still run the risk of losing all of their trout stock? Explain your answer. e Will the size of the trout population ever reach and increase beyond the initial number of 800 fish? Give reasons for your answer. 4 On an island in the Pacific Ocean the population of a species of insect (species A) is increasing geometrically with a population of in 990 and an annual growth rate of.0%. Another species of insect (species B) is also increasing its population, but arithmetically with numbers in 990 and an annual increment of 000 per annum. a Using a spreadsheet or other method, determine the difference in the numbers of the two species during the last decade of the twentieth century (that is, up to 999). b In what year will the first species be greater in number than the second species, assuming that growth rates remain fixed? A scientist has a mathematical model where the species can cohabit provided that they have equal numbers in the year 000. c If the growth rate in species A is to remain unchanged, what would the annual increment in species B need to be to achieve this? d If the annual increment in species B is to remain unchanged, what would the growth rate in species A need to be to achieve this? 5 a A series is given by the equation S n = n + 3n. Show that the sequence is arithmetic and give the expression for the nth term in the sequence, t n. b A series is given by the equation S n = an + bn. Show that the sequence is arithmetic and give the expression for the nth term in the sequence, t n, in terms of a and b. 6 Maria intends to follow a healthy eating plan. She will reduce her daily kilojoules intake by a constant amount each day over a period of 4 days. The graph below shows Maria s kilojoules intake for the first five days. Kilojoule intake Day 5 6 a What was Maria s kilojoules intake on day? b Determine Maria s kilojoules intake on day 6. Exam tip A common incorrect answer was [Assessment report 007] Chapter 5 Sequences and series 49

39 c Maria s kilojoules intake on the nth day is given by the equation K n = a 50 n. Determine the value of a. d On which day will Maria s daily kilojoules intake be 6750? exam TIp A common incorrect answer was [Assessment report 007] exam TIp A common incorrect answer was Day 3. [Assessment report 007] Maria s brother, Rupert, believes he will benefit by reducing his daily kilojoules intake. His kilojoules intake over 4 days will follow a geometric sequence with a common ratio of On day, Rupert s kilojoules intake was 000. e By what percentage is Rupert s kilojoules intake reduced each day? exam TIp A common incorrect answer was 95% or [Assessment report 007] f Determine Rupert s kilojoules intake on day 3. g Write an equation that gives Rupert s kilojoules intake R n on the nth day. exam TIp A frequent error was to use 00 instead of 000. [Assessment report 007] exam TIp A formula in terms of n was required. [Assessment report 007] h Find the difference between Rupert s kilojoules intake on day 9 and day 0. Write your answer correct to the nearest kilojoule. i Determine Rupert s total kilojoules intake from day 8 to day 4 inclusive. Write your answer correct to the nearest kilojoule. Maria decides to improve her fitness level by cycling each day. The time in minutes M n that Maria cycles on the nth day is modelled by the equation M n + = 0.75M n + 8 where M = 0 j For how many minutes will Maria cycle on day 4? k Show that the time Maria cycles each day does not follow an arithmetic or a geometric sequence. exam TIp Many students calculated and listed S 8, S 9, S 0 etc. and then added the result rather than subtracting the results of two geometric series. Others tried S 4 S 8 rather than S 4 S 7. [Assessment report 007] exam TIp For mark, students had to show two subtractions not having the same result and two quotients not having the same result. Some only showed the calculations and did not state that the results within each pair were not equal. Others merely showed the results of the calculations without indicating where these results came from, or their significance. [Assessment report 007] l For how many minutes will Maria cycle on day? Rupert decides to include both swimming and running in his exercise plan. On day, Rupert swims 00 m and runs 500 m. Each day he will increase the distance he swims and the distance he runs. His swimming distance will increase by 50 m each day. His running distance will increase by % of the distance he ran on the previous day. m On which day will the distance Rupert swims first be greater than the distance he runs? [ VCAA 007] ebookplus Digital doc Test Yourself Chapter 5 exam TIp A method mark was available for setting up an equation to find where the two distances were equal. Many students reached the wrong answer with working out that could not be clearly followed and thus earned no marks. [Assessment report 007] 50 maths Quest Standard General mathematics for the Casio Classpad

40 ebookplus activities Chapter opener Digital doc 0 Quick Questions: Warm up with ten quick questions on sequences and series. (page ) 5A Describing sequences Tutorial We int- 0993: Watch how to find terms in three different sequences, by recognising the pattern and by using an expression. (page 4) 5B Arithmetic sequences Tutorial We6 int-0875: Watch how to find specific terms and a general expression for a term in an arithmetic sequence. (page ) Digital docs Spreadsheet 7: Investigate arithmetic sequences. (page 3) SkillSHEET 5.: Practise using elimination to solve simultaneous equations. (page 4) 5C Arithmetic series Tutorial We7 int- 0994: Watch how to find the sum of the first 0 terms in an arithmetic sequence. (page 6) Digital docs Spreadsheet 7: Investigate arithmetic series. (page 7) WorkSHEET 5.: Determine terms, differences between terms, and sums of terms in arithmetic sequences. (page 8) 5D Geometric sequences Tutorial We 0 int-0877: Watch how to find a general expression for a term in a geometric sequence. (page 30) Digital doc Spreadsheet 7: Investigate geometric sequences. (page 3) 5E Geometric series Digital docs Spreadsheet 7: Investigate geometric series. (page 37) WorkSHEET 5.: Determine terms, differences between terms, and sums of terms in geometric sequences. (page 38) 5F Applications of sequences and series Interactivity Sequences and series int-0806: Consolidate your understanding of sequences and series. (page 38) Tutorial We 3 int-0878: Watch how to apply the model of a geometric sequence to a real-life problem. (page 39) Chapter review Digital doc Test Yourself: Take the end-of-chapter test to test your progress. (page 50) To access ebookplus activities, log on to Chapter 5 Sequences and series 5