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ample 1 For all n 2 1, prove that12+2+32+4%= n(n +1)(2n+1)+㥠|
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Answer» For n = 1, the statement reduces to 12 =1 · 2 · 36and is obviously true.Assuming the statement is true for n = k:1^2 + 2^2 + 3^2 + · · · + k2 =k(k + 1)(2k + 1)/6, (1)we will prove that the statement must be true for n = k + 1:1^2 + 2^2 + 3^2 + · · · + (k + 1)^2 =(k + 1)(k + 2)(2k + 3)/6. (2)The left-hand side of (2) can be written as1^2 + 2^2 + 3^2 + · · · + k^2 + (k + 1)^2.In view of (1), this simplifies to: 1^2 + 2^2 + 3^2 + · · · + k^2+ (k + 1)^2 =k(k + 1)(2k + 1)/6+ (k + 1)^2=k(k + 1)(2k + 1) + 6(k + 1)^2/6=(k + 1)[k(2k + 1) + 6(k + 1)]/6=(k + 1)(2k^2 + 7k + 6)/6=(k + 1)(k + 2)(2k + 3)/6.Thus the left-hand side of (2) is equal to the right-hand side of (2). Thisproves the inductive step. Therefore, by the principle of mathematicalinduction, the given statement is true for every positive integer n Like my answer if you find it useful! |
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