# It Doesn't Telescope

Algebra Level 3

$S= \frac{1^{2}}{10}+\frac{2 \times 1^{2}+2^{2}}{10^{2}}+\frac{3 \times 1^{2}+2 \times 2^{2}+3^{2}}{10^{3}}+\cdots$

The sum $$S$$ defined above is an infinite sum whose $$n^\text{th}$$ term is $\dfrac{n \times 1^{2}+(n-1) \times 2^{2}+(n-2) \times 3^{2} + \cdots + n^{2}}{10^{n}}$ for $$n=1,2,\ldots$$.

If $$S$$ can be expressed in the form $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, find $$a+b$$.

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