Mathematics - Sequence & Series Question with Solution | TestHub

Given,$m$is the A.M. of$l \& n.$

$\Rightarrow m=\frac{l+n}{2} ...\left(1\right)$

$G_1, G_2, G_3$are G.M. between$l \& n.$

$\Rightarrow l, G_1, G_2, G_3, n$are in G.P.

$\Rightarrow n=l{\left(r\right)}^4 \Rightarrow r^4=\frac{n}{l} ...\left(2\right)$

$\Rightarrow G_1=lr, G_2=l{r}^2, G_3=l{r}^3$

$\Rightarrow G_1^4+2G_2^4+G_3^4=l^4{r}^4+2\times l^4{r}^8+l^4\times r^{12}$

$=l^4{r}^4\left[1+2r^4+r^8\right]$

$=l^4\times \frac{n}{l}{\left[1+r^4\right]}^2=l^{3}n\times {\left[1+\frac{n}{l}\right]}^2$

$=l^3\times n\times \frac{{\left(l+n\right)}^2}{l^2}$

$=ln\times {\left(2m\right)}^2$(From (1))

$=4l{m}^{2}n$