Question

Let five geometric means are inserted between $\frac{32}{3}$ and $\frac{243}{2}$ then find sum of all the geometric means.

Answer 1

$\frac{32}{3},\frac{243}{2}a{r}^{n+1}=\frac{243}{2}\frac{32}{3}{r}^{n+1}=\frac{243}{2}{r}^{5+1}=\frac{726}{64}\Rightarrow r=\frac{3}{2}$

$G_1=\frac{32}{3}\times \frac{3}{2}=16G_2=16\times \frac{3}{2}=24G_3=24\times \frac{3}{2}=36G_4=36\times \frac{3}{2}=54G_5=54\times \frac{3}{2}=81\sum G_i=16+24+36+54+81=211$