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 $G^{4}_{1} +2G^{4}_{2}+G^{4}_{3}$ equals.

• A. $4\, l^2mn$
• B. $4\, lm^2n$
• C. $4 \,lmn^2$
• D. $4\, l^2m^2n^2$

Answer 1

Answer: (B)

$m =\frac{\ell+ n }{2} \,2 m =\ell+\ell r ^{4}$
$\begin{matrix}\ell&G_{1}&G_{2}&G_{3}&n&\\ \ell&\ell r&\ell r^{2}&\ell r^{3}&\ell r^{4}&=n\end{matrix}$
$\ell^{4} r^{4}+2 \ell^{4} r^{8}+\ell^{4} r ^{12}$
$\Rightarrow \ell^{4} r^{4}\left(1+2 r^{4}+r^{8}\right)$
$\Rightarrow \ell^{4} r^{4}\left(1+r^{4}\right)^{2}$
$\Rightarrow \ell^{4} r^{4}\left(\frac{2 m}{\ell}\right)^{2}$
$\Rightarrow n \cdot \ell^{3} \frac{4 m^{2}}{\ell^{2}}$
$\Rightarrow 4 \ell m^{2} n$