SECTION 12.3 Geometric Sequences; Geometric Series 883 Historical Feature Sequences are among the oldest objects of mathematical investigation, having been studied for over 3500 years. After the initial steps, however, little progress was made until about 1600. Arithmetic and geometric sequences appear in the Rhind papyrus, a mathematical text containing 85 problems copied around 1650 BC by the Egyptian scribe Ahmes from an earlier work (see Historical Problem 1). Fibonacci ( AD 1220) wrote about problems similar to those found in the Rhind papyrus, leading one to suspect that Fibonacci may have had material available that is now lost. This material would have been in the non-Euclidean Greek tradition of Heron (about AD 75) and Diophantus (about AD 250). One problem, again modified slightly, is still with us in the familiar puzzle rhyme “As I was going to St. Ives . . . ” (see Historical Problem 2). The Rhind papyrus indicates that the Egyptians knew how to add up the terms of an arithmetic or geometric sequence, as did the Babylonians. The rule for summing up a geometric sequence is found in Euclid’s Elements (Book IX, 35, 36), where, like all Euclid’s algebra, it is presented in a geometric form. Investigations of other kinds of sequences began in the 1500s, when algebra became sufficiently developed to handle the more complicated problems. The development of calculus in the 1600s added a powerful new tool, especially for finding the sum of an infinite series, and the subject continues to flourish today. 1. Arithmetic sequence problem from the Rhind papyrus (statement modified slightly for clarity) One hundred loaves of bread are to be divided among five people so that the amounts that they receive form an arithmetic sequence. The first two together receive one-seventh of what the last three receive. How many loaves does each receive? [ Partial answer: First person receives 1 2 3 loaves.] 2. The following old English children’s rhyme resembles one of the Rhind papyrus problems. As I was going to St. Ives I met a man with seven wives Each wife had seven sacks Each sack had seven cats Each cat had seven kits [kittens] Kits, cats, sacks, wives How many were going to St. Ives? (a) Assuming that the speaker and the cat fanciers met by traveling in opposite directions, what is the answer? (b) How many kittens are being transported? (c) Kits, cats, sacks, wives; how many? Fibonacci Historical Problems 12.3 Assess Your Understanding 1. If money is invested at a rate of 5.25% compounded quarterly, what is the interest rate per compounding period? (pp. 345–348) ‘Are You Prepared?’ The answer is given at the end of these exercises. If you get the wrong answer, read the pages listed in red. Skill Building In Problems 9–18, show that each sequence is geometric. Then find the common ratio and list the first four terms. 9. s 4 n n { } { } = 10. s 5 n n { } { } ( ) = − 11. a 3 1 2 n n { } ( ) { } = − 12. b 5 2 n n { } ( ) { } = 13. c 2 4 n n 1 { } { } = − 14. d 3 9 n n { } { } = 15. e 7 n n/4 { } { } = 16. f 3 n n2 { } { } = 17. t 3 2 n n n 1 { } { } = − 18. u 2 3 n n n 1 { } { } = − 2. The nth term of a geometric sequence is . 3. In a(n) sequence, the ratio of successive terms is a constant. 4. If r 1, < the sum of the geometric series ar k k 1 1 ∑ = ∞ − is . 5. Multiple Choice If a series does not converge, it is called a(n) series. (a) arithmetic (b) divergent (c) geometric (d) recursive Concepts and Vocabulary 6. True or False A geometric sequence may be defined recursively. 7. True or False In a geometric sequence, the common ratio is always a positive number. 8. True or False For a geometric sequence with first term a1 and common ratio r, where r r 0, 1, ≠ ≠ the sum of the first nterms is S a r r 1 1 . n n 1 = ⋅ − − 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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