Answer
(a) $-\frac{65}{64}$
(b) $-\frac{255}{256}$
(c) $-\frac{1025}{1024}$
Work Step by Step
$\displaystyle \sum_{n=1}^{∞}5(-\frac{1}{4})^n$
(a) $5(-\frac{1}{4})^1+5(-\frac{1}{4})^2+5(-\frac{1}{4})^3=-\frac{5}{4}+\frac{5}{16}-\frac{5}{64}=\frac{-80+20-5}{64}=-\frac{65}{64}$
(b) $5(-\frac{1}{4})^1+5(-\frac{1}{4})^2+5(-\frac{1}{4})^3+5(-\frac{1}{4})^4=-\frac{5}{4}+\frac{5}{16}-\frac{5}{64}+\frac{5}{256}=\frac{-320+80-20+5}{256}=-\frac{255}{256}$
(c) $5(-\frac{1}{4})^1+5(-\frac{1}{4})^2+5(-\frac{1}{4})^3+5(-\frac{1}{4})^4+5(-\frac{1}{4})^5=-\frac{5}{4}+\frac{5}{16}-\frac{5}{64}+\frac{5}{256}-\frac{5}{1024}=\frac{-1280+320-80+20-5}{256}=-\frac{1025}{1024}$
