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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