Discrete and Foundational Mathematics I quiz 7 Solutions

3 2 now add x3 to both sides of 3 thus 13 23 33

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Unformatted text preview: 13 + 23 + 33 + · · · + (x − 1)3 = . (3) 2 Now add x3 to both sides of (3). Thus 13 + 23 + 33 + · · · + (x − 1)3 + x3 = (x − 1) x 2 = x2 = x2 = x2 2 + x3 2 x−1 2 +x 2 (x − 1) +x 4 2 = = = = which contradicts (2). QED. (x − 1) + 4x 4 x2 x2 − 2x + 1 + 4x 4 x2 2 x + 2x + 1 4 x2 2 (x + 1) 4 2 x (x + 1) , 2 b) Proof by induction on n. 1. Base case. Prove (1) is true for n = 1. Note LHS = 13 = 1 and RHS ·2 = 122 = 1. So LHS = RHS. Done with base case. 2. Induction Hypothesis. Assume (1) is true for n = k . So we a...
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This note was uploaded on 01/31/2014 for the course MATH 187 taught by Professor Holmes during the Fall '08 term at Boise State.

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