5333-Solns-Assignment1-Sp13 - Assignment#1 Section 10 3 Prove that 12 22 n2 = 1 6 n(n 1(2n 1 for all n N Let S be the set of positive integers for which

5333-Solns-Assignment1-Sp13 - Assignment#1 Section 10 3...

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Assignment #1 Section 10 3. Prove that 1 2 + 2 2 + · · · + n 2 = 1 6 n ( n + 1)(2 n + 1) for all n N . Let S be the set of positive integers for which the equation holds. Step 1. 1 S : 1 = 1 2 = 1 6 (1)(2)(3) = 1. Therefore, 1 S . Step 2. Assume that the positive integer k S . That is, assume 1 2 + 2 2 + · · · + k 2 = 1 6 k ( k + 1)(2 k + 1) . Step 3. Prove that k + 1 S . 1 2 + 2 2 + · · · + k 2 + ( k + 1) 2 ? = 1 6 ( k + 1)( k + 2)(2 k + 3) 1 6 k ( k + 1)(2 k + 1) + ( k + 1) 2 ? = 1 6 ( k + 1)( k + 2)(2 k + 3) ( k + 1) 6 ( k (2 k + 1) + 6( k + 1)) ? = 1 6 ( k + 1)( k + 2)(2 k + 3) ( k + 1) 6 ( 2 k 2 + 7 k + 6 ) ? = 1 6 ( k + 1)( k + 2)(2 k + 3) 1 6 ( k + 1)( k + 2)(2 k + 3) = 1 6 ( k + 1)( k + 2)(2 k + 3) Therefore, k + 1 S and S = N . 5. Prove that 1 3 + 2 3 + · · · + n 3 = (1 + 2 + · · · + n ) 2 for all n N . Let S be the set of positive integers for which the equation holds. Step 1. 1 S : 1 = 1 3 = 1 2 = 1. Therefore, 1 S . Step 2. Assume that the positive integer k S . That is, assume 1 3 + 2 3 + · · · + k 3 = (1 + 2 + · · · + k ) 2 . Step 3.Prove thatk+ 1S.(1 + 2 +· · ·+k)2+ (k+ 1)3?=(1 + 2 +· · ·+k+ 1)2 parenrightbigg 2 + ( k + 1) 3 ? = (1 + 2 + · · · + k + 1) 2 ( k 2 + 4 k + 4 ) ? = (1 + 2 + · · · + k + 1) 2 parenrightbigg 2 ? = (1 + 2 + · · · + k + 1) 2
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  • Fall '08
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  • Integers, Natural number, Prime number

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