Question

Find the value of n so that $\frac{a^{n+1}+b^{n+1}}{a^n+b^n }$n may be the geometric mean between a and b.

Answer 1

G. M. of a and b is √ab.
By the given condition, $\frac{a^{n+1}+b^{n+1}}{a^n+b^n }=\sqrt{ab}$
Squaring both sides, we obtain
$\frac{(a^{n+1}+b^{n+1})^2}{(a^n+b^n)^2}=\sqrt{ab}$
⇒ a2n+2 + 2an+1 bn+1 + b2n+2 = (ab) (a2n + 2anbn + b2n)
⇒ a2n+2 + 2an+1 bn+1 b2n+2 = a2n+1 b + 2an+1 bn+1 + ab2n+1
⇒ a2n+2 + b2n+2 = a2n+1 b + ab2n+1
⇒ a2n+2 - a2n+1 b = ab2n+1 - b2n+2
⇒ a2n+1 (a - b) = b2n+1 (a - b)
$⇒(\frac{a }{ b})^{2n + 1}=1 = (\frac{a }{b})^0$
⇒ 2n + 1 = 0
$⇒ n = \frac{- 1 }{ 2}$