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.

Answer 1

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

Fifth term of series is

$a{r}^{5-1}=81$ ........(1)

Second term of series is

$a{r}^{2-1}=24$ .........(2)

Dividing (2) by (1) we get,

$\frac{ar}{a{r}^4}=\frac{24}{81}=\frac{8}{27}$

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

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

Substituting r in (2), we get,

$\Rightarrow a=\frac{24\times 2}{3}=16$

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

$S_8=\frac{[{\left(\frac{3}{2}\right)}^8-1]}{\frac{3}{2}-1}$

$\Rightarrow S_8=\frac{16[3^8-2^8]}{2^7}$

$\Rightarrow S_8=32\left(\frac{6561-256}{256}\right)$

$=\frac{32\times 6305}{256}=\frac{6305}{8}$