\(S=\displaystyle\sum_{n=1}^7\left(\frac14\right)^n\)
\(\displaystyle S=\frac14+\left(\frac14\right)^2+\cdots+\left(\frac14\right)^7\)
\(\displaystyle\frac14S=\left(\frac14\right)^2+\left(\frac14\right)^3+\cdots+\left(\frac14\right)^8\)
\(\displaystyle S-\frac14S=\frac14+\left(\left(\frac14\right)^2-\left(\frac14\right)^2\right)+\cdots+\left(\left(\frac14\right)^7-\left(\frac14\right)^7\right)-\left(\frac14\right)^8\)
\(\displaystyle\frac34S=\frac14-\left(\frac14\right)^8\)
\(\displaystyle S=\frac{\frac14-\left(\frac14\right)^8}{\frac34}=\frac{5461}{16384}\)
