It Doesn't Telescope

Algebra Level 3

S=1210+2×12+22102+3×12+2×22+32103+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 SS defined above is an infinite sum whose nthn^\text{th} term is n×12+(n1)×22+(n2)×32++n210n\dfrac{n \times 1^{2}+(n-1) \times 2^{2}+(n-2) \times 3^{2} + \cdots + n^{2}}{10^{n}} for n=1,2,n=1,2,\ldots .

If SS can be expressed in the form ab\frac{a}{b}, where aa and bb are coprime positive integers, find a+ba+b.


Inspiration.

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