Find the sum of the GP :

Sum of a G.P. series is represented by the formula $S_{n}=a \frac{1-r^{n}}{1-r}$ 

when |r|<1. ‘Sn’ represents the sum of the G.P. series upto nth terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.

Here

a = 1

r = (ratio between the n term and n-1 term) $-\frac{1}{3} \div 1=-\frac{1}{3}$

n terms

$\therefore S_{\mathrm{n}}=1 \times \frac{1-\frac{-1}{3}^{\mathrm{n}}}{1-\frac{1}{3}}$

$\Rightarrow \mathrm{S}_{\mathrm{n}}=\frac{1-\frac{1}{3}^{\mathrm{n}}}{\frac{2}{3}}$

$\therefore \mathrm{S}_{\mathrm{n}}=\frac{3-\frac{1}{3}^{\mathrm{n}-1}}{2}$