Abstract Algebra Theory and Applications - Computer Science ...

32 CHAPTER 1 THE INTEGERS9. Use induction to prove that 1 + 2 + 2 2 + · · · + 2 n = 2 n+1 − 1 for n ∈ N.10. Prove thatfor n ∈ N.12 + 1 6 + · · · + 1n(n + 1) = nn + 111. If x is a nonnegative real number, then show that (1 + x) n − 1 ≥ nx forn = 0, 1, 2, . . ..12. Power Sets. Let X be a set. Define the power set of X, denoted P(X),to be the set of all subsets of X. For example,P({a, b}) = {∅, {a}, {b}, {a, b}}.For every positive integer n, show that a set with exactly n elements has apower set with exactly 2 n elements.13. Prove that the two principles of mathematical induction stated in Section 1.1are equivalent.14. Show that the Principle of Well-Ordering for the natural numbers impliesthat 1 is the smallest natural number. Use this result to show that thePrinciple of Well-Ordering implies the Principle of Mathematical Induction;that is, show that if S ⊂ N such that 1 ∈ S and n + 1 ∈ S whenever n ∈ S,then S = N.15. For each of the following pairs of numbers a and b, calculate gcd(a, b) andfind integers r and s such that gcd(a, b) = ra + sb.(a) 14 and 39(b) 234 and 165(c) 1739 and 9923(d) 471 and 562(e) 23,771 and 19,945(f) −4357 and 375416. Let a and b be nonzero integers. If there exist integers r and s such thatar + bs = 1, show that a and b are relatively prime.17. Fibonacci Numbers. The Fibonacci numbers are1, 1, 2, 3, 5, 8, 13, 21, . . . .We can define them inductively by f 1 = 1, f 2 = 1, and f n+2 = f n+1 + f n forn ∈ N.(a) Prove that f n < 2 n .(b) Prove that f n+1 f n−1 = f 2 n + (−1) n , n ≥ 2.(c) Prove that f n = [(1 + √ 5 ) n − (1 − √ 5 ) n ]/2 n√ 5.(d) Show that lim n→∞ f n /f n+1 = ( √ 5 − 1)/2.