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

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

Option 2 Put one penny on the first square doubling the amount for each square… n = What kind of sequence? r = and a = First three terms of sequence? Formula: S(64)= = 64 total squares Geometric 21 S(n) = a ( r n – 1) / ( r – 1 ) 1 ( 2 64 – 1) / ( 2 – 1 ) $ x 10 19

What did you choose? $ 2,080,000 for arithmetic sequence $ x for geometric sequence Did you guess right?

The Sum to Infinity of a Geometric Sequence

Architect Example Architect designing stained glass window 81m 2

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

Pattern Continues 2/3 of remaining part is red

Pattern Continues 2/3 of remaining part is green and so on and so on…

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

Formula 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 Find the sum to infinity for the sequence What is a? What is r? S ∞ = a=12 r=-1/4 12 / (1 – (-1/4) = 12 / (5/4) = 9.6

Example 2 A sequence is Explain why the sequence is a geometric sequence. Means that each term obtained by multiplying by a common ratio Each term is ¼ of previous term

Example 2 continued A sequence is 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 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 = and r = Sequence: S ∞ = Weight = / (1-.75) = 72

Your turn! Page 117 Exercise 14.3 # 1 a, d #2 b #3 b #4 b