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 = a^b2n+1 - b²ⁿ⁺²
⇒ a²ⁿ⁺¹ (a - b) = b²ⁿ⁺¹ (a - b)
\(⇒(\frac{a }{ b})^{2n + 1}=1 = (\frac{a }{b})^0\)
⇒ 2n + 1 = 0
\(⇒ n = \frac{- 1 }{ 2}\)