Answer
$\dfrac{7448}{15,625}=0.476672$
Work Step by Step
We are given the geometric sum:
$S=\dfrac{1}{5}+\dfrac{3}{25}+\dfrac{9}{125}+...+\dfrac{243}{15,625}$
We rewrite the sum in order to determine the pattern:
$S=\dfrac{1}{5}+\dfrac{3}{5^2}+\dfrac{3^2}{5^3}+...+\dfrac{3^5}{5^6}=S_6=\sum_{k=0}^5 \dfrac{1}{5}\left(\dfrac{3}{5}\right)^k$
We have:
$S_6=a+ar+...+ar^5=\sum_{k=0}^5 ar^k=a\dfrac{1-r^6}{1-r}$
The geometric sequence has the ratio $r=\dfrac{3}{5}$ and the first term $a=\dfrac{1}{5}$.
Compute the sum:
$S_6=\sum_{k=0}^5 \dfrac{1}{5}\left(\dfrac{3}{5}\right)^k=\left(\dfrac{1}{5}\right)\dfrac{1-\left(\frac{3}{5}\right)^6}{1-\frac{3}{5}}=\dfrac{7448}{15,625}=0.476672$
