906 CHAPTER 12 Sequences; Induction; the Binomial Theorem Section You should be able to . . . Examples Review Exercises 12.1 1 List the first several terms of a sequence (p. 855) 1–4 1, 2 2 List the terms of a sequence defined by a recursive formula (p. 858) 5, 6 3, 4 3 Use summation notation (p. 859) 7, 8 5, 6 4 Find the sum of a sequence algebraically and using a graphing utility (p. 861) 9 13, 14 5 Solve annuity and amortization problems using recursive formulas (p. 862) 10, 11 37, 39(a) 12.2 1 Determine whether a sequence is arithmetic (p. 869) 1–3 7–12 2 Find a formula for an arithmetic sequence (p. 870) 4, 5 17, 19–21, 38(a) 3 Find the sum of an arithmetic sequence (p. 871) 6–8 7, 10, 14, 38(b), 39 12.3 1 Determine whether a sequence is geometric (p. 875) 1–3 7–12 2 Find a formula for a geometric sequence (p. 877) 4 11, 18, 40(a)–(c), 42, 43(b) 3 Find the sum of a geometric sequence (p. 878) 5, 6 9, 11, 15, 16 4 Determine whether a geometric series converges or diverges (p. 879) 7–9 22–25, 40(d) 5 Solve annuity problems using formulas (p. 881) 10, 11 37 12.4 1 Find the Limit of a Sequence (p. 887) 1–4 26, 27 2 Define Infinite Series and Geometric Series (p. 889) 28 12.5 1 Prove statements using mathematical induction (p. 894) 1–4 30–32 12.6 1 Evaluate ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j (p. 899) 1 33 2 Use the Binomial Theorem (p. 900) 2–5 34–37 Objectives Review Exercises In Problems 1–4, list the first five terms of each sequence. 1. ( ) { } { } ( ) = − + + a n n 1 3 2 n n 2. { } { } = c n 2 n n 2 3. = = − a a a 3; 2 3 n n 1 1 4. = = − − a a a 2; 2 n n 1 1 5. Expand ∑( ) + = k4 2 . k 1 4 6. Express − + − + + 1 1 2 1 3 1 4 1 13 using summation notation. In Problems 7–12, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. 7. { } { } = + a n 5 n 8. { } { } = c n2 n 3 9. { } { } = s 2 n n3 10. 0, 4, 8, 12, . . . 11. 3, 3 2 , 3 4 , 3 8 , 3 16 , . . . 12. 2 3 , 3 4 , 4 5 , 5 6 , . . . In Problems 13–16, find each sum. 13. ∑( ) + = k 2 k 1 30 2 14. ∑( ) − + = k2 8 k 1 40 15. ∑( ) = 1 3 k k 1 7 16. ∑( ) − = 2 k k 1 10 In Problems 17–19, find the indicated term in each sequence. [Hint: Find the general term first.] 17. 9th term of 3, 7, 11, 15, . . . 18. 11th term of 1, 1 10 , 1 100 , . . . 19. 9th term of 2, 2 2, 3 2, . . . In Problems 20 and 21, find a general formula for each arithmetic sequence. 20. 7th term is 31; 20th term is 96 21. 10th term is 0; 18th term is 8
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