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# Ans-9 - Answers Assignment#9 Exercises 1.1 1 Let S be the...

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Answers, Assignment #9 Exercises 1.1. 1. Let S be the set of integers for which the statement is true. Since 1 = 1 · 2 2 = 1, 1 S. Assume that the statement is true for some positive integer n = k Then, for n = k + 1, 1 + 2 + · · · + k + ( k + 1) = [1 + 2 + · · · + k ] + ( k + 1) = k ( k + 1) 2 + ( k + 1) = k ( k + 1) + 2( k + 1) 2 = ( k + 1)( k + 2) 2 Thus, k + 1 S and the statement is true for all n N . 2. Let S be the set of integers for which the statement is true. Since 1 2 = 1 = 1 · 2 · 3 6 = 6 6 = 1, 1 S. Assume that the statement is true for some positive integer n = k Then, for n = k + 1, 1 2 + 2 2 + · · · + k 2 + ( k + 1) 2 = [1 2 + 2 2 + · · · + k 2 ] + ( k + 1) 2 = k ( k + 1)(2 k + 1) 6 + ( k + 1) 2 = k ( k + 1)(2 k + 1) + 6( k + 1) 2 6 = ( k + 1)( k + 2)(2[ k + 1] + 1) 6 Thus, k + 1 S and the statement is true for all n N . 3. Let S be the set of integers for which the statement is true. Since 1 = 1 - r 1 - r = 1, 1 S. Assume that the statement is true for some positive integer n = k Then, for n = k + 1, 1 + r + · · · + r k + r k +1 = [1 + r + · · · + r k ] + r k +1 = 1 - r k +1 1 - r + r k +1 = 1 - r k +1 + (1 - r ) r k +1 1 - r = 1 - r k +2 1 - r Thus, k + 1 S and the statement is true for all n N .

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Ans-9 - Answers Assignment#9 Exercises 1.1 1 Let S be the...

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