Question

If $m$ is the $A.M.$ of two distinct real numbers $l$ and $n$ $(l,n>1)$ and $G_1,G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4+2G_2^4+G_3^4$ equals.

• A. $4l^{2}mn$
• B. $4l{m}^{2}n$
• C. $4lm{n}^2$
• D. $4l^2{m}^2{n}^2$

Answer 1

Answer: (B)

$m=\frac{\ell +n}{2}2m=\ell +\ell r^4$
$\begin{array}{cccccc}\ell & G_1& G_2& G_3& n& \\ \ell & \ell r& \ell r^2& \ell r^3& \ell r^4& =n\end{array}$
${\ell }^4{r}^4+2{\ell }^4{r}^8+{\ell }^4{r}^{12}$
$\Rightarrow {\ell }^4{r}^4\left(1+2r^4+r^8\right)$
$\Rightarrow {\ell }^4{r}^4{\left(1+r^4\right)}^2$
$\Rightarrow {\ell }^4{r}^4{\left(\frac{2m}{\ell }\right)}^2$
$\Rightarrow n\cdot {\ell }^3\frac{4m^2}{{\ell }^2}$
$\Rightarrow 4\ell m^{2}n$