The fifth term of a G.P. is 81 whereas its second term is 24.
Question:

The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.

Solution:

Let a be the first term and r be the common ratio of the G.P.

$a_{2}=24$

$\Rightarrow a r^{2-1}=24$

$\Rightarrow a r=24$   ….(i)

Similarly, $a_{5}=81$

$\Rightarrow a r^{5-1}=24$

$\Rightarrow a r^{4}=81$

$\Rightarrow \frac{24 \times r^{4}}{r}=81$     [From (i)]

$\Rightarrow r^{3}=\frac{81}{24}$

$\therefore r^{3}=\frac{27}{8}$

$\Rightarrow r=\frac{3}{2}$

Putting $r=\frac{3}{2}$ in (i)

$3 a=48$

$\Rightarrow \mathrm{a}=16$

So, the geometric series is $16+24+36+\ldots+16\left(\frac{3}{2}\right)^{8}$

And, $S_{8}=16\left(\frac{\left(\frac{3}{2}\right)^{8}-1}{\frac{3}{2}-1}\right)$

$\Rightarrow S_{8}=32\left(\frac{6561-256}{256}\right)=\frac{32 \times 6305}{256}=\frac{6305}{8}$