# Power Sums: Patterns

Calculus Level 5

All power sums have a closed polynomial forms for integral powers. For example,

$1^2+2^2+3^2+\cdots+n^2=\displaystyle \sum_{k=1}^n k^2=\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}$

More generally

$1^m+2^m+3^m+\cdots+n^m=\displaystyle \sum_{k=1}^n k^m=\displaystyle \sum_{i=1}^{m+1} a_i n^i$

In the case of $$m=2$$, $$a_1=\frac{1}{6}$$, $$a_2=\frac{1}{2}$$, and $$a_3=\frac{1}{3}$$.

When $$m=2017$$, find $$a_{2017}-a_{2015}$$.

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