The first term of a G.P is $27$ and the eight term is $\dfrac{1}{81}$. Find the sum of the first five terms.
First term of G.P $= a = 27$ $\;\;\; \cdots \; (1)$
Let common ratio of G.P $= r$
$n^{th}$ term of G.P $= t_n = ar^{n-1}$
Eight term of G.P $= t_8 = ar^7 = \dfrac{1}{81}$ $\;\;\; \cdots \; (2)$
Dividing equations $(1)$ and $(2)$ we get,
$\dfrac{ar^7}{a} = \dfrac{1}{81 \times 27}$
i.e. $\;$ $r^7 = \left(\dfrac{1}{3}\right)^7$ $\implies$ $r = \dfrac{1}{3}$
Sum of $n$ terms of G.P $= S_n = \dfrac{a \left(1 - r^n\right)}{1 - r}$, $\;$ when $\;$ $r <0$
$\therefore \;$ Sum of first five terms $= S_5 = \dfrac{27 \left[1 - \left(\dfrac{1}{3}\right)^5\right]}{1 - \dfrac{1}{3}}$
i.e. $\;$ $S_5 = \dfrac{27 \times 242 \times 3}{243 \times 2} = \dfrac{121}{3}$