1072 CHAPTER 11 Further Topics in Algebra Use the following guidelines in the exercises below. Summary Exercises on Sequences and Series Arithmetic and Geometric Sequences Given a sequence a1, a2, a3, a4, a5, c , • If the differences a2 - a1, a3 - a2, a4 - a3, a5 - a4, c are all equal to the same number d, then the sequence is arithmetic, and d is the common difference. • If the ratios a2 a1 , a3 a2 , a4 a3 , a5 a4 , c are all equal to the same number r, then the sequence is geometric, and r is the common ratio. Determine whether each sequence is arithmetic or geometric. Then find an and S10. 11. 3, 6, 12, 24, 48, c 12. 2, 6, 10, 14, 18, c 13. 4, 5 2 , 1, - 1 2 , -2, c 14. 3 2 , 1, 2 3 , 4 9 , 8 27 , c 15. 3, -6, 12, -24, 48, c 16. -5, -8, -11, -14, -17, c Evaluate each sum that converges. Identify any that diverge. 17. a ∞ i=1 1 3 1-22i-1 18. a 4 j=1 2 a 1 10b j-1 19. a25 i=1 14 - 6i2 20. a 6 i=1 3i 21. a ∞ i=1 4 a- 1 2b i 22. a ∞ i=1 13i - 22 23. a12 j=1 12j - 12 24. a ∞ k=1 5-k 25. a ∞ i=1 1.0001i 26. Write 0.333c as an infinite geometric series. Find the sum. Determine whether each sequence is arithmetic, geometric, or neither. If it is arithmetic, give the common difference d. If it is geometric, give the common ratio r. 9. 1, 9, 10, 19, 29, c 10. -1, 25, -5, 525, -25, c 5. 3 4 , 1, 4 3 , 16 9 , 64 27 , c 6. 4, -12, 36, -108, 324, c 7. 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , c 8. 5, 2, -1, -4, -7, c 1. 2, 4, 8, 16, 32, c 2. 1, 4, 7, 10, 13, c 3. 3, 1 2 , -2, - 9 2 , -7, c 4. 1, 1, 2, 3, 5, 8, c
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