Find the value of n, if [(a^{n+1} + b^{n+1})/ (a^n +b^n)] is the arithmetic mean between a and b.

Question

Find the value of n, if  \(\frac{a^{n+1} +b^{n+1}}{a^n +b^n}\) is the arithmetic mean between a and b.

Answer 1

The arithmetic mean between a and b is \(\frac{a+b}{2}. \)

Given ,

\(\frac{a+b}{2} \) = \(\frac{a^{n+1} + b^{n+1}}{a^n +b^n}\) 

⇒ (aⁿ + bⁿ) (a + b) = 2(aⁿ⁺¹ + bⁿ⁺¹) ⇒ aⁿ⁺¹ + aⁿb + bⁿa + bⁿ⁺¹ = 2aⁿ⁺¹ + 2bⁿ⁺¹

⇒ aⁿ⁺¹ + bⁿ⁺¹ = aⁿb + bⁿa ⇒ (aⁿ⁺¹ – baⁿ) – (bⁿa – bⁿ⁺¹) = 0

⇒ aⁿ (a – b) – bⁿ (a – b) = 0 ⇒ (aⁿ – bⁿ) (a – b) = 0

⇒ (aⁿ – bⁿ) = 0 or (a – b) = 0

But a ≠ b ⇒ a – b ≠ 0 

⇒ n = 0