135-a3 - x k-1-6 x k-2 for k 3 . Prove that for all n P , x...

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Assignment 3 – Questions 1. Use induction to prove that 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + ··· + 1 2 n 1 + n 2 . [This can be used to prove that the infinite harmonic series X r =1 1 r diverges.] 2. Let a 1 = 2 and a n +1 = 5 a n - 4 a n for n 1. Prove that for all n 1, 1 a n a n +1 4 . 3. A sequence of integers x 1 , x 2 , x 3 ,... is defined by x 1 = 3, x 2 = 7, and x k = 5
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Unformatted text preview: x k-1-6 x k-2 for k 3 . Prove that for all n P , x n = 2 n + 3 n-1 . 4. Let a = 1 and a 1 = 1 and for n 2 let a n = 2 a n-1 + a n-2 . Prove that for all n 0, a n = 1 2 ( (1 + 2) n + (1- 2) n ) . 1...
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