soln3 - MATH 135 Algebra Solutions to Assignment 3 1(a Let...

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MATH 135 Algebra, Solutions to Assignment 3 1: (a) Let a 1 = 1 and a n +1 = 3 a n + 2 for n 1. Show that a n = 2 · 3 n - 1 - 1 for all n 1. Solution: We claim that a n = 2 · 3 n - 1 - 1 for all n 1. When n = 1 we have a n = a 1 = 1 and 2 · 3 n - 1 - 1 = 2 · 3 0 - 1 = 2 · 1 - 1 = 1 so the claim is true when n = 1. Let k 1 and suppose the claim is true when n = k , that is suppose that a k = 2 · 3 k - 1 - 1. Then when n = k + 1 we have a n = a k +1 = 3 a k + 2 = 3 ( 2 · 3 k - 1 - 1 ) + 2 = 2 · 3 k - 3 + 2 = 2 · 3 k - 1 = 2 · 3 n - 1 - 1 . Thus the claim is true when n = k + 1. By Mathematical Induction, a n = 2 · 3 n - 1 - 1 for all n 1. (b) Let a 1 = 3 and a n +1 = 2 a n - 1 for n 1. Find a closed form formula for a n . Solution: Using the given recursion formula, we find that a 1 = 3, a 2 = 5, a 3 = 9, a 4 = 17 and a 5 = 33. Notice that a n = 2 n + 1 for n = 1 , 2 , 3 , 4 , 5. We claim that a n = 2 n + 1 for all n 1. When n = 1 the claim is true. Let k 1 and suppose that the claim is true when n = k , that is suppose that a k = 2 k + 1. Then when n = k + 1 we have a n = a k +1 = 2 · a k - 1 = 2(2 k + 1) - 1 = 2 k +1 + 2 - 1 = 2 k +1 + 1 = 2 n + 1 . Thus the claim is true when n = k + 1. By Mathematical Induction, we have a n = 2 n + 1 for all n 1. (c) Let a 1 = 2 and a n +1 = 5 a n - 4 a n for n 1. Show that 1 a n a n +1 4 for all n 1. Solution: We claim that 1 a n a n +1 4 for all n 1. We have a 1 = 2 and the recursion formula gives a 2 = 5 a 1 - 4 a 1 = 5 · 2 - 4 2 = 3, and so we do have 1 a 1 a 2 4 and so the claim is true when n = 1. Let k 1 and suppose the claim is true when n = k , that is suppose that 1 a k a k +1 4. We have 1 a k a k +1 4 = 1 1 a k 1 a k +1 1 4 = 4 4 a k 4 a k +1 1 = ⇒ - 4 ≤ - 4 a k ≤ - 4 a k +1 ≤ - 1 = 1 5 - 4 a k 5 - 4 a k +1 4 = 1 5 a k - 4 a k 5 a k +1 - 4 a k +1 4 = 1 a k +1 a k +2 4 . Thus the claim is true when n = k + 1. By Mathematical Induction, 1 a n a n +1 4 for all n 1. 2: (a) Let a 0 = 0 and a 1 = 1 and for n 2 let a n = a n - 1 +6 a n - 2 . Show that a n = 1 5 ( 3 n - ( - 2) n ) for all n 0.
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