# For what value of n is $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$ the GM of 4 and 16.

$\begin{array}{1 1} 4 \\ \large\frac{-1}{4} \\ \frac{-1}{2} \\ 2 \end{array}$

Answer : (c) $\frac{-1}{2}$
Explanation :
$\frac{4^{n+1}+16^{n+1}}{4^n+16^n}=\sqrt{4*16}=\sqrt{64}=8$
$4^{n+1}+16^{n+1}=8(4^n+16^n)$
$4^n(8-4)=16^n(16-8)$
$4^{n+1}=16^n\;.8$
$4^{n+1}=4^{2n}\;.4\;.4^{1/2}$
$n+1=2n+1+1/2$
$n=\frac{-1}{2}.$