Find the sum 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 $r=\frac{0.015}{0.15}=0.1$

$\mathrm{S}_{\mathrm{n}}=\frac{\mathrm{a}\left(1-\mathrm{r}^{\mathrm{n}}\right)}{1-\mathrm{r}}$

$\therefore S_{20}=\frac{0.15\left[1-(0.1)^{20}\right]}{1-0.1}$

$=\frac{0.15}{0.9}\left[1-(0.1)^{20}\right]$

$=\frac{15}{90}\left[1-(0.1)^{20}\right]$

$=\frac{1}{6}\left[1-(0.1)^{20}\right]$