Linear Algebra: A Modern Introduction, 3rd Edition - Appendix B ...

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Exercises B In Exercises 1–8, use mathematical induction to prove the given formula for the specified values of n. 1. 1  5  9  p  14n  32  2n2  n for all n  1. 2. 2  5  8  p  13n  12  n  1.

3. 12  22  32  p  n2 

n13n  12 for all 2

n1n  12 12n  12 for all 6

n  1. 4. 1 # 2  2 # 3  3 # 4  p  n1n  12  n1n  12 1n  22 for all n  1. 3 5. 1  2  4  8  p  2n  2n1  1 for all n  0. rn1  1 6. For r  1, 1  r  r2  p  rn  for all r1 n  0. 7. 1 # 1!  2 # 2!  p  n # n!  1n  12! 1 for all n  1. (The factorial function is n!  n1n12 1n22 p 2 # 1.) 8. 12  22  32  42  p  112 n1 n2  112 n1n1n  12 for all n  1. 2 In Exercises 9–24, use mathematical induction to prove the given statement. 9. n2  n is even for all n  0. 10. n3n is divisible by 3 for all n  0. 11. 5n1 is divisible by 4 for all n  0. 12. 32n2n is divisible by 7 for all n  0.

13. 2n 7 n2 for all n  5. 14. 3n 7 n2 for all n  0. 15. 1  14  19  p  n12  2  n1 for all n  1. 16. Let x be a real number greater than 1. Prove that 11  x2 n  1  nx for all n  0. (This inequality is called Bernoulli’s Identity.) 17. Let a and b be nonzero real numbers. Prove that 1ab 2 n  anbn for all n  0.

18. Let a be a nonzero real number. Prove that 1am 2 n  amn for all m, n  0.

19. xn  1 is divisible by x  1 for all n  1. 20. xn  yn is divisible by x  y for all n  1. 21. A set with n elements has exactly 2n subsets. 22. Given n distinct points in a plane such that no three of the points are on a straight line, there exist exactly n2n line segments joining pairs of points. 2 23. The sum of the interior angles of a convex polygon with n sides is 1n22180 degrees. (A polygon is convex if every line segment joining two vertices lies inside the polygon.) 24. The maximum number of regions into which a circle can be subdivided by n straight lines is 1n2  n  22>2. (See the Remark after Example B.6.) 25. Make a conjecture about the value of

1

1

 #  1#2 2 3

1 1 p for n  1. Then prove that your 3#4 n1n  12 conjecture is correct using mathematical induction. 1

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26. Make a conjecture about the value of a 1 

1 b  22

1 1 a 1  2 b  p  a 1  2 b for n  2. Then 3 n prove that your conjecture is correct using mathematical induction. In Exercises 27–30, use the second principle of mathematical induction to prove the given statement. 27. Every integer n  1 can be factored in the form n  2km for some integer k  0 and some odd integer m. 28. Every integer n  1 can be written as a sum of distinct powers of 2. 29. Every integer n  8 can be written in the form n  3a  5b for some nonnegative integers a and b. 30. Every integer n  12 can be written in the form n  3a  7b for some nonnegative integers a and b. In Section 4.6, the Fibonacci sequence 0, 1, 1, 2, 3, 5, ... is defined by f0  0, f1  1 and, for n  2, fn  fn1  fn2. In Exercises 31–36, prove the given property of the Fibonacci sequence using an appropriate form of mathematical induction. n

31. a fi  fn21 for all n  0. i0 n

32. a f2i1  f2n2 for all n  0. i0 n

33. a f i2  fn fn1 for all n  0. i0

34. fn is even if and only if n is divisible by 3. 35. fm1 fn  fm fn1  fmn for all m  1, n  0 [Hint: Fix m and perform induction on n.] 36. fm divides fmn for all m, n  0 [Hint: Fix m and perform induction on n. Use Exercise 35.] 37. A 2n  2n chessboard has one square removed. Prove by mathematical induction that, for all n  1, such a chessboard can be tiled by L-shaped tiles formed from three 1  1 tiles, as shown below.

38. Using mathematical induction, prove that for all n  2, 41  21 11 p is an irrational number, where n is the number of square root symbols. (For example, when n  3, we have 41  21 11.) [Hint: See Exercise A.51.] The Tower of Hanoi puzzle was introduced in 1883 by the French mathematician Edouard Lucas. It consists of three pegs affixed to a base and a tower of n disks of different diameters stacked on one of the pegs. The goal is to transfer the tower of disks to a different peg by moving only one disk at a time and never placing a disk on top of a smaller disk.

39. Using mathematical induction, prove that the minimum number of moves needed to transfer a tower of n disks is 2n1. 40. Suppose the pegs are arranged in a straight line and the tower initially starts on one of the end pegs. Suppose further that each disk must move either to or from the center peg. Prove that the minimum number of moves needed to transfer a tower of n disks from one end peg to the other end peg is 3n1.

In Exercises 41 and 42, find the flaw in the “proof.” 41. 1  2 “Proof” We prove that for all positive integers n, n  n  1. Assume that for some k  1, we have k  k  1. Then k  1  1k  12  1. So n  n  1 for all n  1, by the principle of mathematical induction. Setting n  1, it follows that 1  2. 42. 1  2 “Proof ” We prove that in any nonempty set of n positive integers, all the integers are equal. Clearly in a set with only one integer a, a  a. Now assume that in any set of k positive integers, all k integers are equal. Consider a set 5a1, a2, p, ak1 6 with k  1 positive

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integers. If we remove ak1, then in the resulting set 5a1, a2, p, ak 6 we have a1  a2  p  ak, by the induction hypothesis. On the other hand, if we remove a1, then in the resulting set 5a2, p, ak1 6 we have a2  p  ak1, by the induction hypothesis.

Hence, a1  a2  p  ak1 and so all elements of 5a1, a2, p, ak1 6 are equal. By the principle of mathematical induction, it follows that in any nonempty set of n positive integers, all the integers are equal. In particular, in 51, 26, we have 1  2.

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