Question

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

Answer 1

$T_5=a{r}^4=81$, $T_2=ar=24$
Dividing, $r^3=\frac{81}{24}=\frac{27}{8}$
$\therefore
{ r }=\dfrac { 3 }{ 2 } >1$andhence$a=16$.
$\therefore S_8=\frac{a(r^n-1)}{r-1}=\frac{16}{3/2-1}\left[\right(3/2{)}^8-1]$
$=32.\frac{(3^8-2^8)}{2^8}=\frac{1}{8}(3^4-2^4)(3^4+2^4)=\frac{1}{8}(65\times 97)$.