1086 CHAPTER 11 Further Topics in Algebra Write out in full and verify the statements S1, S2, S3, S4, and S5 for the following. Then use mathematical induction to prove that each statement is true for every positive integer n. See Example 1. 7. 1 + 3 + 5 + g+ 12n - 12 = n2 8. 2 + 4 + 6 + g+ 2n = n1n + 12 Let Sn represent the given statement. Show that Sn is true for the natural numbers n specified. See Example 3. 27. 2n 72n, for all n such that n Ú 3 28. 3n 72n + 1, for all n such that n Ú 2 29. 2n 7n2, for all n such that n Ú 5 30. 4n 7n4, for all n such that n Ú 5 31. n! 72n, for all n such that n Ú 4 32. n! 73n, for all n such that n Ú 7 Let Sn represent the given statement, and use mathematical induction to prove that Sn is true for every positive integer n. See Example 1. Follow these steps. (a) Verify S1. (b) Write Sk. (c) Write Sk+1. (d) Assume that Sk is true and use algebra to change Sk toSk+1. (e) Write a conclusion based on Steps (a)–(d). 9. 3 + 6 + 9 + g+ 3n = 3n1n + 12 2 10. 5 + 10 + 15 + g+ 5n = 5n1n + 12 2 11. 2 + 4 + 8 + g+ 2n = 2n+1 - 2 12. 3 + 9 + 27 + g+ 3n = 1 213n+1 - 32 13. 12 + 22 + 32 + g+ n2 = n1n + 1212n + 12 6 14. 13 + 23 + 33 + g+ n3 = n21n + 122 4 15. 5 # 6 + 5 # 62 + 5 # 63 + g+ 5 # 6n = 616n - 12 16. 7 # 8 + 7 # 82 + 7 # 83 + g+ 7 # 8n = 818n - 12 17. 1 1 # 2 + 1 2 # 3 + 1 3 # 4 + g+ 1 n1n + 12 = n n + 1 18. 1 1 # 4 + 1 4 # 7 + 1 7 # 10 + g+ 1 13n - 2213n + 12 = n 3n + 1 19. 1 2 + 1 22 + 1 23 + g+ 1 2n = 1 - 1 2n 20. 4 5 + 4 52 + 4 53 + g+ 4 5n = 1 - 1 5n Prove each of the following for every positive integer n. Use steps (a)–(e) as in Exercises 9–20. See Example 2. 21. If a 71, then an 71. 22. If a 71, then an 7an-1. 23. If 0 6a 61, then an 6an-1. 24. The bionomial 1x - y2 is a factor of x2n - y2n. 25. 1am2n = amn (Assume a and m are constant.) 26. 1ab2n = anbn (Assume a and b are constant.) Solve each problem. 33. Number of Handshakes Suppose that each of the n1for n Ú 22 people in a room shakes hands with everyone else, but not with himself or herself. Show that the number of handshakes is n2 - n 2 . An infinite ladder illustrates the concept of mathematical induction.
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