Question

The fifth term of a G.P. is 81 whereas its second term is 24. Let the sum of its first eight terms be $\frac{1}{k}(65\times 97).$ Find $k$ ?

Answer 1

$T_5=a{r}^4=81$ and $T_2=ar=24$

Dividing $r^3=\frac{81}{24}=\frac{27}{8}$

$r=\frac{3}{2}>1$ and hence $a=16$

$s_8=\frac{a(r^n-1)}{r-1}=\frac{16}{3/2-1}\left[\right(3/2{)}^8-1]=32\cdot \frac{(3^8-2^8)}{2^8}=\frac{1}{8}(3^4-2^4\left)\right(3^4+2^4)=\frac{1}{8}(65\times 97).$