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S=1×2210+2×32102+3×42103+4×52104+⋯S=\dfrac{1×2^2}{10}+\dfrac{2×3^2}{10^2}+\dfrac{3×4^2}{10^3}+\dfrac{4×5^2}{10^4}+\cdotsS=101×22+1022×32+1033×42+1044×52+⋯
If SSS is in the form AB\dfrac{A}{B}BA, where AAA and BBB are coprime positive integers, find the value of A+BA+BA+B.
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