The Sum to Infinity of a geometric sequence

The Sum to Infinity of a geometric sequence Sequences and Series Unit

The Chess Problem The king of Loolooland was under attack by bandits in the Looloo Forest, and was rescued by the brave Sir Lagbehind, a knight of the Rhomboid Table. The king was so grateful that he promised the knight a great reward.

The plot thickens “See this chessboard, ” the king said, pulling a chessboard out of his voluminous traveling robes, “I’m going to give you this chessboard. But before I do, I’m going to put money on each square of the board. You get to decide what I put on the squares. ”

Dollars? “I can put $1000 on the first square adding a thousand for each square…

Cents? … or I can put one penny on the first square doubling the amount for each square…

Would you choose 1 or 2? 1. 2. Put $1000 on the first square adding another thousand for each square… Put one penny on the first square doubling the amount for each square… *A chess board has 8 x 8 squares

Option 1 n n Put $1000 on the first square adding another thousand for each square… n = 64 a = 1000 d = 1000 What kind of sequence? Arithmetic First three terms of sequence? <1000, 2000, 3000, … > n n Formula: S(n) = (n/2)[2 a + (n-1)d] S(64) = (64/2)[2(1000) + (64 -1)(1000)] = 32 [ 2000 + 63000] = $ 2, 080, 000

Option 2 n Put one penny on the first square doubling the amount for each square… n = 64 total squares What kind of sequence? Geometric r =2 and a = 1 First three terms of sequence? <1, 2, 4, …> n Formula: n S(64) n n S(n) = a ( r n – 1) / ( r – 1 ) = 1 ( 2 64 – 1) / ( 2 – 1 ) = $ 1. 845 x 10 19

What did you choose? n n $ 2, 080, 000 for arithmetic sequence $ 1. 845 x 10 19 for geometric sequence Did you guess right?

The Sum to Infinity of a Geometric Sequence

Architect Example n n Architect designing stained glass window 81 m 2

Pattern n n 2/3 of the window is blue How many squares have been colored so far? 54

Pattern Continues n n 2/3 of remaining part is red 54 + 18

Pattern Continues n n 2/3 of remaining part is green and so on… 54 + 18 + 6

Sequence <54, 18, 6, …> What is the next number of the sequence? 2 Explain why this sequence is geometric. What is the common ratio? r = 2/3 Can this process continue indefinitely?

Definition n When common ratio is between – 1 and 1 -1 < r < 1 n n the terms become smaller in size as we continue along the sequence Process called ‘taking partial sums’

Formula n n The total of all the terms is called the sum to infinity Formula for the sum to infinity of a geometric sequence is: S∞ = a / (1 – r) (for – 1 < r < 1 only)

Example n n n Find the sum to infinity for the sequence <12, -3, ¾, -3/16, …> What is a? What is r? a=12 r=-1/4 S∞ = 12 / (1 – (-1/4) = 12 / (5/4) = 9. 6

Example 2 n A sequence is <96, 24, 6, …> Each term is ¼ of previous term n Explain why the sequence is a geometric sequence. Means that each term obtained by multiplying by a common ratio

Example 2 continued n n A sequence is <96, 24, 6, …> Show why the sum of these terms cannot exceed 128 S∞ = a / (1 -r) = 96 / (1 – ¼) = 96 / (3/4) = 96 x (4/3) = 128

Application n n A person who weighs 104 kg plans to lose 8 kg during the first three months of a diet, 6 kg during the next 3 months, 4. 5 kg during the next three month and so on. What is the persons weight after a large number of years?

Solution A person who weighs 104 kg plans to lose 8 kg during the first three months of a diet, 6 kg during the next 3 months, 4. 5 kg during the next three month and so on. a= 8 and r= <8, 6, 4. 5, …> S∞ = 8 / (1 -. 75) S∞ = 32 Weight = 104 - 32 = 72 Sequence: . 75

Your turn! n n n Page 117 Exercise 14. 3 # 1 a, d #2 b #3 b #4 b