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The closed form sum of \(12 \left[ 1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right]\)for n>= 1 is n(n+1)(n+2)(an+b). Find an+b.

 Feb 15, 2020
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Using the formulas 1 + 2 + ... + n = n(n + 1)/2 and 1^2 + 2^2 + ... + n^2 = n(n + 1)(2n + 1)/6, the sum in the problem works out to 2n^4 + 9n^3 + 13n^2 + 6n = n(n + 1)(n + 2)(2n + 3), so an + b = 2n + 3.

 Feb 15, 2020

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