The $2^{\text {nd }}$ and $5^{\text {th }}$ terms of a GP are $\frac{-1}{2}$ and $\frac{1}{16}$ respectively. Find the sum of $n$ terms GP up to 8 terms.
$2^{\text {nd }}$ term $=a r^{2-1}=a r^{1}$
$5^{\text {th }}$ term $=a r^{5-1}=a r^{4}$
Dividing the $5^{\text {th }}$ term using the $3^{\text {rd }}$ term
$\frac{a r^{4}}{a r}=\frac{\frac{1}{16}}{\frac{-1}{2}}$
$r^{3}=-\frac{1}{8}$
$\therefore r=-\frac{-1}{2}$
∴ a = 1
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.
n = 8 terms
$\mathrm{S}_{\mathrm{n}}=1 \times \frac{1-\frac{-1^{8}}{2}}{1-\frac{-1}{2}}$
$\Rightarrow \mathrm{S}_{\mathrm{n}}=\frac{1-\frac{1}{256}}{\frac{3}{2}}$
$\Rightarrow \mathrm{S}_{\mathrm{n}}=\frac{\frac{255}{256}}{\frac{3}{2}}$
$\therefore \mathrm{S}_{\mathrm{n}}=\frac{170}{256}$