# Powers of 1/a

Algebra Level 3

$S = 1 + 2\left(\frac{1}{5}\right) + 3\left(\frac{1}{5}\right)^2 + 4\left(\frac{1}{5}\right)^3 \ldots .$

If $S = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers, what is the value of $a+b$?

Details and assumptions

$S$ is obtained as the sum of infinitely many terms. The $n$th term is $n\left(\frac{1}{5}\right)^{n-1}$.

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