Question

Find the sum of $20$ terms of a G.P $0.15,0.015,\mathrm{0.0015....}$

Answer 1

Given, $0.15,0.015,\mathrm{0.0015...}$ are in G.P.
Here, first term $=a=0.15$ and
common ratio $=r=\frac{0.015}{0.15}=\frac{\frac{15}{1000}}{\frac{15}{100}}=0.1<1$

We know, the sum of $n$ terms of $G.P.$ is
$S_n=\frac{a(1-r^n)}{1-r}$

For sum of $20$ terms, put $n=20$
$S_{20}=\frac{0.15(1-(0.1{)}^{20})}{1-0.1}$
$\Rightarrow S_{20}=\frac{0.15(1-(0.1{)}^{20})}{0.9}$
$\Rightarrow S_{20}=\frac{15(1-(0.1{)}^{20})}{90}$
$\therefore S_{20}=\frac{1}{6}[1-(0.1{)}^{20}]$