PATTERNS, SEQUENCES & SERIES (LIVE) 07 APRIL 2015 Section A: Summary Notes and Examples

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1 PATTERNS, SEQUENCES & SERIES (LIVE) 07 APRIL 05 Section A: Summary Notes and Examples Grade Revision Before you begin working with grade patterns, sequences and series, it is important to revise what you learnt in grade about quadratic sequences. A quadratic sequence is a sequence in which the second difference is constant. The general term of this sequence is T n = an + bn + c a = is a constant term which is equal to half the second difference b = constant term c = constant term Example Consider the pattern: 5; ; 7; 0;. Write down the next two terms. Determine an expression for the n th terms. Show that the sequence will never have a term with a value less than Solutions. ; 0. Begin by identifying the sequence. Since the sequence doesn t have a common first difference or a constant ratio, we check to see if the sequence is quadratic. d = a = To find b and c substitute n = into T n = an + bn + c Equation T = + b + c 5 = b + 4 = b + c Now substitute n = Equation T = + ()b + c = + b + c 6 = b + c Now solve equation and simultaneously Equation minus equation 0 = b 4 = 0 + c 4 = c T n = n 0n + 4 Page

2 . n 0n + 4 < n 0n + 5 < 0 (n 5) < 0 This is not true for any values of n thus the sequence will not have a term less than Arithmetic Sequences and Series An arithmetic sequence or series is a linear number pattern in which the first difference is constant. The general term formula allows you to determine any specific term of an arithmetic sequence. And the sum of formula determines the sum of a specific number of terms of an arithmetic series. The formulae are as follows: T n = a + n d S n = n [a + n d] where a = first term and d = constant difference where a = first term and d = constant difference S n = n [a + l] where l is the last term Note: d = T T T = a T = a + d T = a + d etc. Example The 9 th term of an arithmetic sequence is, while the st term is 5. (a) Determine the first three terms of the sequence. T 9 = a + 8d = T = a + 0d = d = 6 d = a + 8 a = 0 = 0; 9 ; 9 (b) Which term of the sequence is equal to 9? T n = 9 T n = a + (n )d 0 + n = 9 n = 49 n = 98 n = 99 T 99 = 9 Page

3 Example Given: (a) Calculate the sum of the given series a = 8 d = 8 S 80 = 80 n = = 90 = 90 (b) Hence calculate the sum of the following series: = [a = d = T n = 90] + n = 90 + n = 80 n = 80 S 80 = = = 845 Geometric Sequences and Series A geometric sequence or series is an exponential number pattern in which the ratio is constant. The general term formula allows you to determine any specific term of a geometric sequence. You have also learnt formulae to determine the sum of a specific number of terms of a geometric series. The formulae are as follows: T n = ar n S n = a(rn ) r r = T T where r T = a T = ar T = ar etc. Page

4 Example In a geometric sequence in which all terms are positive, the sixth term is and the eighth term is 7. Determine the first term and constant ratio. T 6 = and T 8 = 7 ar 5 = ar 7 = 7 ar7 ar 5 = 7 r = 7 r = 9 r = r = (terms are positive) a( ) 5 = a = ( ) 5 a = ( ) 4 a = ( )4 a = 9 Convergent Geometric Series Consider the following geometric series: We can work out the sum of progressive terms as follows: S = = 0,5 S = + 4 = 4 = 0,75 S = = 7 8 = 0,875 (Start by adding in the first term) (Then add the first two terms) (Then add the first three terms) S 4 = = 5 = 0,975 (Then add the first four terms) If we continue adding progressive terms, it is clear that the decimal obtained is getting closer and closer to. The series is said to converge to. The number to which the series converges is called the sum to infinity of the series. There is a useful formula to help us calculate the sum to infinity of a convergent geometric series. The formula is S = a r If we consider the previous series Page 4

5 It is clear that a = and r = S = a r S = = A geometric series will converge only if the constant ratio is a number between negative one and positive one. In other words, the sum to infinity for a given geometric series will exist only if < r < If the constant ratio lies outside this interval, then the series will not converge. For example, the geometric series will not converge since the sum of the progressive terms of the series diverges because r = which lies outside the interval < r < Example Given the geometric series: 8x + 4x + x 4 + (a) Determine the n th term of the series. T n = ar n T n = (8x ) x n (b) For what value(s) of x will the series converge? < x < = < x < (c) Calculate the sum of the series to infinity if x = S = a r S = 8x x S = 8( ) ( ) S = 7 Sigma Notation 6 Sigma means sum of, for example n= n + means the sum of the five terms in the sequence n+. We determine the number of terms in this sequence by subtracting the number at the bottom,, from the number at the top, 6, and as seen below. There are 5 terms in the sequence. 6 n= 6 n= n + = [6 + ] n + = = 5 Page 5

6 Example (a) Calculate the value of 00 k= 00 k= (k ) k = [ 00 ] = From the question we can see that the sequence is arithmetic and further more we have the last term therefore, we can use the formula S n = n [a + l] to calculate the sum: S 00 = 00 [ + 99] S 00 = 0000 (b) Write the following series in sigma notation: The series is arithmetic. There are also 6 terms in the series. a = d = n = 6 We can determine the general term as follows: T n = a + n d T n = + n () T n = + n T n = n We can now write the series in sigma notation as follows: 6 n= (k ) Section B: Practice Questions Question Consider the sequence ; ; 8; ; 8; ; 8; ; 8;. Determine the 00 th term. (). Determine the sum of the first 00 terms. () Question The th and 7 th terms of an arithmetic sequence are 5 and 5 respectively.. Which term of the sequence is equal to (6) Page 6

7 Question In a geometric sequence, the 6 th term is 4 and the rd term is 7. Determine:. The constant ratio. (4). The sum of the first 0 terms. (4) Question 4 Consider the sequence: ; 4; ; 7; ; 0; If the pattern continues in the same way, write down the next TWO terms in the sequence. () 4. Calculate the sum of the first 50 terms of the sequence. (7) Question 5 5. Determine n if n r = 6r = Prove that: n k n = n 4n Question 6 k= (6) Consider the series n= ( x)n 6. For which values of x will the series converge? () 6. If x =, calculate the sum to infinity of this series. () Question 7 A sequence of squares, each having side, is drawn as shown below. The first square is shaded, and the length of the side of each shaded square is half the length of the side of the shaded square in the previous diagram. (7) DIAGRAM DIAGRAM DIAGRAM DIAGRAM 4 7. Determine the area of the unshaded region in DIAGRAM 7. () 7. What is the sum of the areas of the unshaded regions on the first seven squares? (5) Page 7

8 Question 8 A plant grows,5 m in st year. Its growth each year thereafter, is of its growth in the previous year. 8. What is the greatest height it can reach? () Section C: Solutions. T n = a + n d T n = a + n d. T 00 = = 49 S n = n [a + n d] S 00 = 00 [( ) + 00 (5)] S 00 = 4550 T 00 = 49 S n = n [a + n d] T () (). T = 5 T 7 = 5 a + d = 5 a + 6d = 5 a + d = 5. A a + 6d = 5. B 6d = 6 A B d = 6 a + ( 6) = 5 a = 7 = 5 a = 87 a + d = 5 a + 6d = 5 d = 6 a = n + 6 = n = 9 (6) T n = a + n d = 87 + n 6 = 87 6n + 6 = n = 9 T 9 =. T 6 = 4 AND T = 7 a. r 5 = 4 A a. r = 7 B a. r 5 = 4 A a. r = 7 B r = 7 8. A B r = a. r 5 = 4 a. r = 7 r = 7 8 r = (4). Substitute r = into A a 5 = 4 a 5 = 4 a = a = S 0 = ( )0 S 0 = ( )0 answer (4) S 0 = ( )0 = 66,565 Page 8

9 4. 6 ; answers () 4. S 50 = 5 terms of st sequence which is geometric +5 terms of nd sequence which is arithmetic. S 50 = to 5 terms to 5 terms S 50 = ( ) () S 50 = 0, separating into an arithmetic and geometric series ( )5 correct formulae () answer (7) S 50 = 00,00 5. n r 6r = n = 456 = n = 456 This is an arithmetic sequence since we can see that d = 6 S n = n (a + n d) expanding correct formula 456 = n (a + n d 0 = n² + n 456 n + 8 n = = n (a + n d 456 = n ( 5 + n 6 n = 8 n = or n = (7) 456 = n (0 + 6n 6) 456 = n (4 + 6n) 456 = n + n² 0 = n² + n 456 n + 8 n = 0 n = 8 or n = n = 8 or n = n = 5. n k= k n = 5n + 7n + 9n + + n n expanding a = 5n, d = n a = 5n, d = n and number of terms = n number of terms = n Page 9

10 S n = n [a + n d] S n = n [(5n) + n (n ] S n = n [0n + n 6n] S n = n [n + 4n] correct formula substitution answer (6) S n = n n + n²(n ) S n = n² 4n + n³ n² = n³ 4n 6. n ( x) n = ( x) + ( x) + ( x) + ( x)4 + r = x < x < < x < () = x + x² + x³ + x⁴ The series converges for < x < < x < 6. a = r = S = = 4 = 4 a and r S formula () DIAGRAM DIAGRAM DIAGRAM DIAGRAM 4 7. Area of unshaded square = Area of large square Area of small shaded square = 4 4 = 6 = () Page 0

11 7. Sum of the unshaded areas of the first seven squares: = = Getting the pattern for the unshaded areas correct formula substitution answer (5) = 7 ( 4 )7 4 = 7,595 = 5, S =,5 S = 45m Thus the greatest height is 4,5 m correct formula substitution answer Page