Find the sum up to $20$ terms in the geometric progression $0.15,0.015,0.0015........$
Find the sum up to $20$ terms in the geometric progression $0.15,0.015,0.0015........$
The given $G.P.$ is $0.15,0.015,0.00015 \ldots$
Here, $a=0.15$ and $r=\frac{0.015}{0.15}=0.1$
$S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}$
$\therefore S_{20}=\frac{0.15\left[1-(0.1)^{20}\right]}{1-0.1}$
$=\frac{0.15}{0.9}\left[1-(0.1)^{20}\right]$
$=\frac{15}{90}\left[1-(0.1)^{20}\right]$
$=\frac{1}{6}\left[1-(0.1)^{20}\right]$
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