Question

Determine the sum \[\sum_{k=1}^{\infty} \frac{4^k +3^{k-1}-2^{k-2}}{5^{k+1}}\]

Answer 1

I don't think the series above either geometric nor arithmetic.

Answer 2

Hint: It is a sum of three Geometric series. Try splitting the sum up like this

\[\frac{1}{5}\sum_{k=1}^{\infty}(\frac{4}{5})^{k}+ \frac{1}{15}\sum_{k=1}^{\infty}(\frac{3}{5})^{k}-\frac{1}{20}\sum_{k=1}^{\infty}(\frac{2}{5})^{k}\]