UNCORRECTED. Geometric sequences. The rule that we use to get from one number to the next is of the form. t n t n 1. = r

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1 4 Geometric sequences Objectives To recognise geometric sequences, and to find their terms, recurrence relations and numbers of terms. To calculate the sum of the terms in a geometric series. To calculate the sum of the terms in an infinite geometric series. To investigate applications of geometric sequences. In Chapter 4, we began our study of sequences of numbers by looking at arithmetic sequences. For an arithmetic sequence, the rule that we use to get from one number to the next is of the form t n = t n + d or equivalently t n t n = d where d is a constant. We say that there is a common difference. In this chapter, we study geometric sequences. For example: a, 2, 4, 8,... b 0., 0.0, 0.00,... c, 2, 4, 8,... d 2,, 2, 4,... The rule that we use to get from one number to the next is of the form t n = rt n or equivalently t n t n = r where r is a constant. We say that there is a common ratio. In Chapter 4, we applied arithmetic sequences to simple interest. In this chapter, we investigate several applications of geometric sequences, including compound interest, depreciation, annuities and loans. Note: Geometric sequences can be explored using the spreadsheet application of your calculator; instructions are given in the appendices of the Interactive Textbook. Knowledge check See the online test of required knowledge, with links to revision lessons. Chapter 4

2 482 Chapter 4: Geometric sequences 4A Geometric sequences Recall that the numbers of a sequence are called its terms. The nth term of a sequence is denoted by the symbol t n. So the first term is t, the 2th term is t 2, and so on. A sequence in which each successive term is found by multiplying the previous term by a fixed amount is called a geometric sequence. That is, a geometric sequence has a recurrence relation of the form t n = rt n where r is a constant. For example: 2, 6, 8, 54,... is a geometric sequence. The nth term of a geometric sequence is given by t n = t r n where r is the common ratio of successive terms, that is, r =, for all k >. We often t k denote the first term by a and write this formula as t n = ar n Note: In a geometric sequence, the nth term t n is an exponential function of n. Example Find the 0th term of the sequence 2, 6, 8,.... a = 2, r = 3 t n = ar n t 0 = Example 2 = For a geometric sequence, the first term is 8 and the fourth term is 44. Find the common ratio. a = 8, t 4 = 44 t 4 = 8 r 4 = 44 8r 3 = 44 r 3 = 8 r = 2 The common ratio is 2. t k

3 4A Geometric sequences 483 Example 3 For a geometric sequence 36, 8, 9,..., the nth term is 9. Find the value of n. 6 a = 36, r = 2 ( ) n t n = 36 = ( ) n = ( ) n = 2 64 ( ) n ( 6 = 2 2) Example 4 n = 6 n = 7 The third term of a geometric sequence is 0 and the sixth term is 80. Find the common ratio and the first term. t 3 = ar 2 = 0 () t 6 = ar 5 = 80 (2) Divide (2) by (): ar 5 ar = r 3 = 8 r = 2 Substitute in (): a 4 = 0 a = 5 2 The common ratio is 2 and the first term is 5 2.

4 484 Chapter 4: Geometric sequences Example 5 Georgina draws a pattern consisting of a number of equilateral triangles. The first triangle has sides of length 4 cm and the side length of each successive triangle is one and a half times the side length of the previous one. a What is the side length of the fifth triangle? b Which triangle has a side length of cm? a a = 4, r = 3 b a = 4, r = 3 2 2, t n = t n = ar n ( 3 4 t 5 = ar 4 = 4 2) = 20 4 The fifth triangle has a side length of 20 4 cm. Geometric mean t n = ar n = ( 3 ) n 4 = ( 3 ) n = 729 ( = 2 Therefore n = 6 and so n = 7. The seventh triangle has a side length of cm. In Chapter 4, we considered the arithmetic mean a + b of two numbers a and b, and the 2 relationship with arithmetic sequences. The geometric mean of two positive numbers a and b is defined as ab. If positive numbers a, c, b are consecutive terms of a geometric sequence, then c a = b c Section summary c = ab A geometric sequence has a recurrence relation of the form t n = rt n, where r is a constant. Each successive term is found by multiplying the previous term by a fixed amount. For example: 2, 6, 8, 54,... The nth term of a geometric sequence is given by t n = ar n where a is the first term and r is the common ratio of successive terms, that is, t k r =, for all k >. t k ) 6

5 4A 4A Geometric sequences 485 Example Example 2 Example 3 Example 4 Exercise 4A For a geometric sequence t n = ar n, find the first four terms given that: 2 a a = 3, r = 2 b a = 3, r = 2 c a = 0 000, r = 0. d a = r = 3 Find the specified term in each of the following geometric sequences: 5 a 2, 0, 50,... find t 7 b 7, 5 7, 5 2,... find t 6 c, 4, 6,... find t 5 d a x, a x+, a x+2,... find t 6 3 Find the rule for the geometric sequence whose first few terms are: 4 5 a, 4, 6 b 3, 2, 4 3 Find the common ratio for the following geometric sequences: a the first term is 2 and the sixth term is 486 b the first term is 25 and the fifth term is 6 25 c 2, 4, 8, 6 A geometric sequence has first term and common ratio 2. Which term of the 4 sequence is 64? 6 If t n is the nth term of the following geometric sequences, find n in each case: 7 a 2, 6, 8,... t n = 486 b 5, 0, 20,... t n = 280 c 768, 384, 92,... t n = 3 e 4 3, 2 3, 3,... t n = 96 d 8 9, 4 3, 2,... t n = 27 4 The 2th term of a geometric sequence is 2 and the 5th term is 54. Find the 7th term. 8 A geometric sequence has t 2 = 8 and t 4 = 9. Find the two possible values for t 7. 9 The number of fish in the breeding tanks of a fish farm follow a geometric sequence. The third tank contains 96 fish and the sixth tank contains 768 fish. Example 5 0 a How many fish are in the first tank? b How many fish are in the 0th tank? An algal bloom is growing in a lake. The area it covers triples each day. a If it initially covers an area of 0 m 2, what is the area it will cover after one week? b If the lake has a total area of m 2, how long before the entire lake is covered? A ball is dropped from a height of 2 m and continues to bounce so that it rebounds to three-quarters of the height from which it previously falls. Find the height it rises to on the fifth bounce. SF

6 486 Chapter 4: Geometric sequences 4A 2 An art collector has a painting that is increasing in value by 8% each year. If the painting is currently valued at $2500: a How much will it be worth in 0 years? b How many years before its value exceeds $00 000? 3 The Tour de Moravia is a cycling event which lasts for 5 days. On the first day the cyclists must ride 20 km, and each successive day they ride 90% of the distance of the previous day. a How far do they ride on the 8th day? b On which day do they ride 30.5 km? 4 A child negotiates a new pocket-money deal with her unsuspecting father in which she receives cent on the first day of the month, 2 cents on the second, 4 cents on the third, 8 cents on the fourth, and so on, until the end of the month. How much would the child receive on the 30th day of the month? (Give your answer to the nearest thousand dollars.) 5 The first three terms of a geometric sequence are 4, 8, 6. Find the first term in the sequence which exceeds The first three terms of a geometric sequence are 3, 9, 27. Find the first term in the sequence which exceeds The number of type A apple bugs present in an orchard is estimated to be , and the number is reducing by 50% each week. At the same time it is estimated that there are 40 type B apple bugs, whose number is doubling each week. After how many weeks will there be the same number of each type of bug? 8 Find the geometric mean of: a 5 and 720 b and 6.25 c and 3 d x 2 y 3 and x 6 y 3 9 Consider the geometric sequence, a, a 2, a 3,.... Suppose that the sum of two consecutive terms in the sequence gives the next term in the sequence. Find a. 20 A bottle contains 000 ml of pure ethanol. Then 300 ml is removed and the bottle is topped up with pure water. The mixture is stirred. a What is the volume of ethanol in the bottle if this process is repeated five times in total? b How many times should the process be repeated for there to be less than ml of ethanol in the bottle? 2 The rectangle shown has side lengths a and b. a Find the side length of a square with the same perimeter. Comment. b Find the side length of a square with the same area. Comment. b a SF CU

7 4B Geometric series 487 4B Geometric series The sum of the terms in a geometric sequence is called a geometric series. The symbol S n is used to denote the sum of the first n terms of a sequence. That is, S n = a + ar + ar ar n We can find a formula for S n as follows. Let Then Subtract () from (2): Therefore S n = a + ar + ar ar n () rs n = ar + ar 2 + ar ar n (2) rs n S n = ar n a S n (r ) = a(r n ) S n = a(rn ) r For values of r such that < r <, it is often more convenient to use the equivalent formula S n = a( rn ) r which is obtained by multiplying both the numerator and the denominator by. Example 6 Find the sum of the first nine terms of the sequence 3, 9, 27, 8,... a = 3, r = 3, n = 9 S n = a( rn ) r ( 3( ) 9 ) 3 S 9 = 3 = ( ( 9 ) 2 3)

8 488 Chapter 4: Geometric sequences Example 7 For the geometric sequence, 3, 9,..., find how many terms must be added together to obtain a sum of 093. a =, r = 3, S n = 093 S n = a(rn ) = 093 r (3 n ) = n = n = 287 A calculator can be used to find n = 7. Seven terms are required to give a sum of 093. Example 8 In the 5-day Tour de Moravia, the cyclists must ride 20 km on the first day, and each successive day they ride 90% of the distance of the previous day. a How far do they ride in total to the nearest kilometre? b After how many days will they have ridden half that distance? a a = 20, r = 0.9 b a = 20, r = 0.9, S n = S n = a( rn ) r S 5 = 20( (0.9) 5) 0.9 = They ride 953 km. S n = a( rn ) = r 20 ( (0.9) n) = n = n = n = n = A calculator can be used to find n 4.8. Thus they pass the halfway mark on the fifth day.

9 4B 4B Geometric series 489 Section summary Example 6 Example 7 Example 8 The sum of the first n terms of a geometric sequence is given by S n = a + ar + ar ar n S n = a(rn ) r Exercise 4B or S n = a( rn ) r Find the specified sum for each of the following geometric series: a find S 0 b find S 6 c find S 9 2 For the geometric sequence 2, 6, 8,..., write an expression for: a the nth term b the sum of the first n terms. 3 For the geometric sequence, 2,,..., write an expression for: 4 a the nth term b the sum of the first n terms. 4 Find: 5 a b c For the geometric sequence 3, 6, 2,..., find how many terms must be added together to obtain a sum of For the geometric sequence 24, 2, 6,..., find how many terms must be added together to obtain a sum of Gerry owns a milking cow. On the first day he milks the cow, it produces 600 ml of milk. On each successive day, the amount of milk increases by 0%. a How much milk does the cow produce on the seventh day? b How much milk does it produce in the first week? c After how many days will it have produced a total in excess of ml? 8 On Monday, William spends 20 minutes playing the piano. On Tuesday, he spends 25 minutes playing, and on each successive day he increases the time he spends playing in the same ratio. a For how many minutes does he play on Friday? b How many minutes in total does he play from Monday to Friday? c On which day of the following week will his total time playing pass 5 hours? SF

10 490 Chapter 4: Geometric sequences 4B 9 A ball dropped from a height of 5 m rebounds from the ground to a height of 0 m. With each successive rebound, it rises to two-thirds of the height of the previous rebound. What total distance will it have travelled when it strikes the ground for the 0th time? 0 An insurance broker makes $5 000 commission on sales in her first year. Each year, she increases her sales by 5%. a How much commission would she make in her fifth year? b How much commission would she make in total over 5 years? For a geometric sequence with nth term t n : a if t 3 = 20 and t 6 = 60, find S 5 b if t 3 = 2 and t 8 = 8, find S 8. 2 a How many terms of the geometric sequence where t =, t 2 = 2, t 3 = 4,... must be taken for S n = 255? b Let S n = n. Find the values of n for which S n > Find x 2 + x 4 x 6 + x 8 + x 2m, where m is even. 4 A sheet of A4 paper is about 0.05 mm thick. The paper is torn in half, and each half is again torn in half, and this process is repeated for a total of 40 times. a How high will the stack of paper be if the pieces are placed one on top of the other? b How many times would the process have to be repeated for the stack to first reach the moon, km away? 5 Which would you prefer: $ million given to you every week for a year, or c in the first week, 2c in the second, 4c in the third, and so on, until the end of the year? 4C Applications of geometric sequences Compound interest One application of geometric sequences is compound interest. Compound interest is interest calculated at regular intervals on the total of the amount originally invested and the amount accumulated in the previous years. For example, assume that $000 is invested at 0% per annum, compounded annually. Then the value of the investment increases by 0% each year. After year, the investment will have grown to $000. = $00. 2 After 2 years, the investment will have grown to $00. = $ = $20. 3 After 3 years, the investment will have grown to $20. = $ = $33. The value of the investment after n years will be $000. n. SF CU

11 4C Applications of geometric sequences 49 Compound interest Suppose that $P is invested at an interest rate of R% per annum, compounded annually. Then the value of the investment after n years, $A n, is given by A n = Pr n, where r = + R 00 Example 9 Marta invests $2500 at 7% p.a. compounded annually. a Find the value of her investment after 5 years. b Find how long it takes until her investment is worth $ The value after n years is A n = Pr n, where P = 2500 and r =.07. a A 5 = Pr 5 b A n = Pr n = = 2500(.07) 5 = The value of the investment after 5 years is $ (.07) n = n = 4 log 0 (.07 n ) = log 0 4 n log 0 (.07) = log 0 4 n = log 0 4 log 0 (.07) n By the end of the 2st year, the investment will be worth over $ Note: For part b, the number of years can also be found by trial and error or by using a graphics calculator. Compound interest using a recurrence relation Example 9 can also be solved using a spreadsheet. Let $A n be the value of the investment at the end of the nth year. These values can be found recursively by A 0 = 2500 and A n =.07 A n To find the values using a spreadsheet: In cell A, enter the value 0. In cell A2, enter the formula = A +. In cell B, enter the initial value In cell B2, enter the formula =.07 B. Fill down in columns A and B. At the end of year 5, the amount is $ By filling down further in columns A and B, you can find the year that the amount reaches $ Year A Amount B

12 492 Chapter 4: Geometric sequences Depreciation Depreciation occurs when the value of an asset reduces as time passes. For example, suppose that a new car is bought for $20 000, and that its value depreciates by 0% each year. After year, the car s value will have fallen to $ = $ After 2 years, the car s value will have fallen to $ = $ After 3 years, the car s value will have fallen to $ = $ The value of the car after n years will be $ n. Depreciation Suppose that an asset has initial value $P and that its value depreciates at a rate of R% per annum. Then the value of the asset after n years, $D n, is given by D n = Pr n, where r = R 00 Example 0 A machine bought for $5 000 depreciates at the rate of 2 2 % per annum. a What will be the value of the machine after 9 years? b After how many years will its value drop below 0% of its original cost? The value after n years is D n = Pr n, where P = and r = 0.25 = a D 9 = Pr 9 = (0.875) 9 = The value of the machine after 9 years is $ Regular payments b We want to find the smallest value of n for which D n < 0.P. D n = Pr n < 0.P (0.875) n < n < 0. A calculator gives the solution n > The value will drop below 0% of the original cost during the 8th year. We now look at situations where compound interest is combined with equal payments at regular intervals of time. Examples include superannuation contributions, loan repayments and annuities. We will focus on yearly payments, but regular payments are also often made weekly, monthly or quarterly. In the following two examples, we find the solutions using geometric series. The solutions can also be found using a spreadsheet.

13 4C Applications of geometric sequences 493 Regular deposits into a savings account Example Sophie plans to retire in 5 years. She decides to deposit $6000 into a bank account at the start of each year until her retirement. The interest rate is 6% p.a. compounded annually. What will be the account balance when Sophie retires at the end of the 5th year? The first $6000 is in the bank for 5 years and so contributes 6000(.06) 5. The second $6000 is in the bank for 4 years and so contributes 6000(.06) 4. The third $6000 is in the bank for 3 years and so contributes 6000(.06) 3.. The final $6000 is in the bank for one year and so contributes 6000(.06). The final amount in the account is 6000(.06) (.06) (.06) (.06) This is a geometric series with a = 6000(.06), r =.06 and n = 5. S 5 = a(r5 ) = 6000(.06)(.06 5 ) = r.06 Sophie will have $ in her bank account. She has contributed $ In general, if $P is deposited at the start of each year into an account earning compound interest of R% p.a., then the account balance after n years, $A n, is given by A n = Pr(rn ), where r = + R r 00 Loan repayments Example 2 Luke borrows $ and undertakes to repay $6000 at the end of each year. Interest of 0% p.a. is charged on the unpaid debt. a How much does he owe after the 8th repayment? b How long does it take to pay off the loan? Let $D n be the amount still owing after the nth repayment. a Amount owing after the st repayment: D = Amount owing after the 2nd repayment: D 2 = D = ( ) = ( +.)

14 494 Chapter 4: Geometric sequences Amount owing after the 3rd repayment: D 3 = D = ( ( +.) ) = ( ) Following this pattern, the amount owing after the 8th repayment is D 8 = ( ) (. = ) = (. 8 ) = After the 8th repayment, he owes $ b After the nth repayment, the amount owing is D n = n (. n ) = n We want to find when the debt is zero: n = 0. n = 6 Using a calculator gives n 8.8. It takes 9 years to pay off the loan. In general, if $P is borrowed at an interest rate of R% p.a. and a repayment of $Q is made at the end of each year, then the amount owing after n years, $D n, is given by D n = Pr n Q(rn ), where r = + R r 00 Loan repayments using a recurrence relation Example 2 can be solved using a spreadsheet. Let $D n be the amount owing at the end of the nth year. Then D 0 = and D n =. D n 6000 To find the values using a spreadsheet: In cell A, enter the value 0. In cell A2, enter the formula = A +. In cell B, enter the initial value In cell B2, enter the formula =. B Fill down in columns A and B. After 8 years, the debt is $ By filling down further, you can find the year that the debt reaches zero. Year A Owing B

15 4C 4C Applications of geometric sequences 495 Section summary Example 9 Example 0 Compound interest Suppose that $P is invested at an interest rate of R% per annum, compounded annually. Then the value of the investment after n years, $A n, is given by A n = Pr n, where r = + R 00 Depreciation Suppose that an asset has initial value $P and that its value depreciates at a rate of R% per annum. Then the value of the asset after n years, $D n, is given by D n = Pr n, where r = R 00 Situations involving regular payments can be investigated by: finding a pattern in the calculation of the first few values and then using a geometric series finding the recurrence relation for the values and using a spreadsheet. Exercise 4C $5000 is invested at 6% p.a. compounded annually. a Find the value of the investment after 6 years. b Find how long it will take for the original investment to double in value. 2 How much would need to be invested at 8.5% p.a., compounded annually, to yield a return of $8000 after 2 years? 3 The profits of a cosmetics company have been increasing by 5% per annum since its formation. The profit in the first year was $ a Find a formula for the profit in the nth year. b In which year did the annual profit first exceed $ ? c Find a formula for the total profit over the first n years. A car bought for $ depreciates at the rate of 5% per annum. a What will be the value of the car after 3 years? b After how many years will its value drop below 50% of its original cost? 5 What annual compound interest rate would be required to triple the value of an investment of $200 in 0 years? 6 The value of a car is $ when new. If its value depreciates by 5% each year, after how many years will its value be less than $0 000? SF

16 496 Chapter 4: Geometric sequences 4C Example Example 2 7 At the beginning of each year, an investor deposits $ into a fund that pays 5% p.a. compounded annually. How much is the investment worth after 0 years? 8 I wish to accumulate $ over 20 years at 0% p.a. compounded annually. What should be the amount of my annual payments? 9 Chen pays $ into an investment fund at the start of each year, and the fund earns compound interest at a rate of 6% p.a. 0 a How much is the investment worth at the end of 0 years? b After how many years will the value of the investment be over $ ? Daniel borrows $ and undertakes to repay $0 000 at the end of each year. Interest of 5% p.a. is charged on the unpaid debt. a How much does he owe after the 0th repayment? b How long does it take to pay off the loan? Grace lends $ on the condition that she is repaid the money in 5 equal yearly installments. If she receives interest at the rate of 4% p.a., what is the amount of each installment? 2 Andrew invests $000 at 20% simple interest for 0 years. Bianca invests her $000 at 2.5% compound interest for 0 years. At the end of 0 years, whose investment is worth more? 3 By sampling, it is estimated that there are trout in a lake. Assume that the trout population, left untouched, would increase by 5% per annum. It is known that 2000 trout per year are removed by fishing. a How many trout are there in the lake after: i year ii 2 years iii 3 years? b Write a recurrence relation that gives the number of trout in the lake after n years in terms of the number of trout in the lake after n years. c Write a formula for the number of trout in the lake after n years in terms of n. d Find the number of trout in the lake after 5 years. 4 When Emma retired from work at the start of January, she invested a lump sum of $ at an interest rate of 0% p.a. compounded annually. She now uses this account to pay herself an annuity of $ at the end of December every year. a What is the amount left in the account at the end of: i the first year ii the second year iii the third year? b Write a recurrence relation that gives the account balance after n years in terms of the account balance after n years. c Write a formula for the account balance after n years in terms of n. d For how many years will Emma be able to pay herself an annuity of $ before the account balance becomes too low? SF CU

17 4D Zeno s paradox and infinite geometric series 497 4D Zeno s paradox and infinite geometric series A runner wants to go from point A to point B. To do this, he would first have to run half the distance, then half the remaining distance, then half the remaining distance, and so on. A 2 The Greek philosopher Zeno of Elea, who lived about 450 BC, argued that since the runner has to complete an infinite number of stages to get from A to B, he cannot do this in a finite amount of time, and so he cannot reach B. In this section we see how to resolve this paradox. Infinite geometric series If a geometric sequence has a common ratio with magnitude less than, that is, if < r <, then each successive term is closer to zero. For example, consider the sequence 3, 9, 27, 8,... In Example 6 we found that the sum of the first 9 terms is S The sum of the first 20 terms is S We might conjecture that, as we add more and more terms of the sequence, the sum will get closer and closer to 0.5, that is, S n 0.5 as n. An infinite series t + t 2 + t 3 + is said to be convergent if the sum of the first n terms, S n, approaches a limiting value as n. This limit is called the sum to infinity of the series. If < r <, then the infinite geometric series a + ar + ar 2 + is convergent and the sum to infinity is given by S = a r Proof We know that S n = a( rn ) = a r r arn r As n, we have r n 0 and so arn r 0. Hence S n a as n. r Resolution of Zeno s paradox Assume that the runner is travelling at a constant speed and that he takes minute to run half the distance from A to B. Then he takes 2 minute to run half the remaining distance, and so on. The total time taken is This is an infinite geometric series, and the formula gives S = a r = = 2. 2 This fits with our common sense: If the runner takes minute to cover half the distance, then he will take 2 minutes to cover the whole distance. 4 8 B

18 498 Chapter 4: Geometric sequences 4D Example 3 Find the sum to infinity of the series a = 2, r = 2 and so S = = 2 Note: This result is illustrated by the unit square shown. Divide the square in two, then divide one of the resulting rectangles in two, and so on. The sum of the areas of the rectangles equals the area of the square. Example A square has a side length of 40 cm. A copy of the square is made so that the area of the copy is 80% of the original. The process is repeated so that each time the area of the new square is 80% of the previous one. If this process is repeated indefinitely, find the total area of all the squares. The area of the first square is 40 2 = 600 cm 2. We have a = 600 and r = 0.8, giving S = 600 = 8000 cm2 0.8 Example 5 Express the recurring decimal as the ratio of two integers = We have a = 0.32 and r = 0.0, giving i.e. S = = = Exercise 4D Skillsheet Find: Example 3 a b

19 4D 4D Zeno s paradox and infinite geometric series 499 Example 4 Example 5 2 An equilateral triangle has perimeter p cm. The midpoints of the sides are joined to form another triangle, and this process is repeated. Find the perimeter and area of the nth triangle, and find the limits as n of the sum of the perimeters and the sum of the areas of the first n triangles. 3 A rocket is launched into the air so that it reaches a height of 200 m in the first second. Each subsequent second it gains 6% less height. Find how high the rocket will climb. 4 A patient has an infection that, if it exceeds a certain level, will kill him. He is given a drug to inhibit the spread of the infection. The drug acts such that each day the level of infection only increases by 65% of the previous day s level. The level of infection on the first day is 450, and the critical level is 280. Will the infection kill him? 5 A man can walk 3 km in the first hour of a journey, but in each succeeding hour walks half the distance covered in the preceding hour. Can he complete a journey of 6 km? Where does this problem cease to be realistic? 6 A frog standing 0 m from the edge of a pond sets out to jump towards it. Its first jump is 2 m, its second jump is 2 m, its third jump is 8 m, and so on. Show that the frog will never reach the edge of the pond. 7 A computer virus acts in such a way that on the first day it blocks out one-third of the area of the screen of an infected computer. Each successive day it blocks out more of the screen: an area one-third of that it blocked the previous day. If this continues indefinitely, what percentage of the screen will eventually be blocked out? 8 A stone is thrown so that it skips across the surface of a lake. If each skip is 30% less than the previous skip, how long should the first skip be so that the total distance travelled by the stone is 40 m? 9 A ball dropped from a height of 5 m rebounds from the ground to a height of 0 m. With each successive rebound it rises two-thirds of the height of the previous rebound. If it continues to bounce indefinitely, what is the total distance it will travel? 0 Express each of the following periodic decimals as the ratio of a pair of integers: a 0. 4 b c 0. 3 d e 0. 9 f 4. The sum of the first four terms of a geometric series is 30 and the sum to infinity is 32. Find the first two terms. 2 Find the third term of a geometric sequence that has a common ratio of and a sum 4 to infinity of 8. 3 Find the common ratio of a geometric sequence with first term 5 and sum to infinity 5. SF 4 For any number x > 2, show that there is an infinite geometric series such that a = 2 and the sum to infinity is x.

20 500 Chapter 4: Geometric sequences Review AS Nrich Chapter summary The nth term of a sequence is denoted by t n. A recurrence relation enables each subsequent term to be found from previous terms. A sequence specified in this way is said to be defined recursively. e.g. t = 3, t n = 2t n A sequence may also be defined by a rule that is stated in terms of n. e.g. t n = 3 2 n Geometric sequences A geometric sequence has a recurrence relation of the form t n = rt n where r is a constant. Each successive term is found by multiplying the previous term by a fixed amount. For example: 2, 6, 8, 54,... The nth term of a geometric sequence is given by t n = ar n where a is the first term and r is the common ratio (i.e. r = for all k > ). t k The geometric mean of two numbers a and b is ab. Geometric series The sum of the terms in a geometric sequence is called a geometric series. For r, the sum of the first n terms of a geometric sequence is given by S n = a(rn ) r or S n = a( rn ) r For < r <, the sum S n approaches a limiting value as n, and the series is said to be convergent. This limit is called the sum to infinity and is given by S = a r Applications of geometric sequences Compound interest Suppose that $P is invested at an interest rate of R% per annum, compounded annually. Then the value of the investment after n years, $A n, is given by A n = Pr n, where r = + R 00 Depreciation Suppose that an asset has initial value $P and that its value depreciates at a rate of R% per annum. Then the value of the asset after n years, $D n, is given by D n = Pr n, where r = R 00 t k

21 Chapter 4 review 50 Technology-free questions Find the first six terms of the following sequences: a t = 3, t n = 2t n b t = 4, t n = 2t n Find the first six terms of the following sequences: a t n = 2 n b t n = 2 n 3 Nick invests $5000 at 5% p.a. compound interest at the beginning of the year. At the beginning of each of the following years, he puts a further $500 into the account. a Write down the amount of money in the account at the end of each of the first two years. b Set up a recurrence relation to generate the sequence for the investment. 4 The 6th term of a geometric sequence is 9 and the 0th term is 729. Find the 4th term. 5 One thousand dollars is invested at 3.5% p.a. compounded annually. Find the value of the investment after n years. 6 The first term of a geometric sequence is 9 and the third term is 4. Find the possible values for the second and fourth terms. 7 The sum of three consecutive terms of a geometric sequence is 24 and the sum of the next three terms is also 24. Find the sum of the first 2 terms. 8 Find the sum of the first eight terms of the geometric sequence with first term 6 and common ratio 3. 9 Find the sum to infinity of The numbers x, x + 4, 2x + 2 are three successive terms of a geometric sequence. Find the value of x. Multiple-choice questions The first three terms of the sequence defined by the rule t n = 2 n are A, 2, 3 B, 2, 4 C, 3, 7 D 2, 4, 8 E 3, 5, 9 2 If t = 3 and t n+ = 2t n + 3, then t 4 is A 9 B 2 C 2 D 27 E 45 3 The sixth term of the geometric sequence 2, 8, 6 3,... is A B C D For the sequence 8, 4, 2,..., we have S 6 = A B C D 5 E E SF SF Review

22 502 Chapter 4: Geometric sequences Review 5 For the sequence 8, 4, 2,..., we have S = A B 0 C 6 D If $2000 is invested at 5.5% p.a. compounded annually, the value of the investment after 6 years is A $ B $ 62.8 C $ D $ E $ If S = 37.5 and r =, then a equals 3 2 A B 2.5 C D 25 E Extended-response questions Each time Lee rinses her hair after washing it, the result is to remove a quantity of shampoo from her hair. With each rinse, the quantity of shampoo removed is one-tenth of that removed by the previous rinse. a If Lee washes out 90 mg of shampoo with the first rinse, how much will she have washed out altogether after six rinses? b How much shampoo do you think was present in her hair at the beginning? 2 A prisoner is trapped in an underground cell, which is inundated by a sudden rush of water. The water comes up to a height of m, which is one-third of the height of the ceiling (3 m). After an hour another inundation occurs, and the water level in the cell rises by 3 m. After a second hour another inundation raises the water level by 9 m. If this process continues for 6 hours, write down: a the amount the water level will rise at the end of the sixth hour b the total height of the water level then. If this process continues, do you think the prisoner, who cannot swim, will drown? Why? 3 After an undetected leak in a storage tank, the staff at an experimental station were subjected to 500 curie hours of radiation the first day, 400 the second day, 320 the third day, and so on. Find the number of curie hours they were subjected to: a on the 4th day b during the first five days of the leak. 4 A rubber ball is dropped from a height of 8 m. Each time it strikes the ground, it rebounds two-thirds of the distance through which it has fallen. a Find the height that the ball reaches after the sixth bounce. b Assuming that the ball continues to bounce indefinitely, find the total distance travelled by the ball. E

23 Chapter 4 review In payment for loyal service to the king, a wise peasant asked to be given one grain of rice for the first square of a chessboard, two grains for the second square, four for the third square, and so on for all 64 squares of the board. The king thought that this seemed fair and readily agreed, but was horrified when the court mathematician informed him how many grains of rice he would have to pay the peasant. How many grains of rice did the king have to pay? (Leave your answer in index form.) 6 The following diagrams show the first four steps in forming the Sierpinski triangle. Step Step 2 Step 3 Step 4 The diagrams are produced in the following way: Step Step 2 Step 3 Step 4 Start with an equilateral triangle of side length unit. Subdivide it into four smaller congruent equilateral triangles and colour the central one blue. Repeat Step 2 with each of the smaller white triangles. Repeat again. a How many white triangles are there in the nth diagram (that is, after Step n)? b What is the side length of a white triangle in the nth diagram? c What fraction of the area of the original triangle is still white in the nth diagram? d Consider what happens as n approaches infinity. 7 The Sierpinski carpet is formed from a unit square in a way similar to the Sierpinski triangle. The following diagrams show the first three steps. Step Step 2 Step 3 a How many white squares are there in the nth diagram (that is, after Step n)? b What is the length of a side of a white square in the nth diagram? c What is the fraction of the area of square which is white in the nth diagram? d Consider what happens as n approaches infinity. CU Review