The value of n, for which \(\dfrac{a^{n+1} + b^{n+1}}{a^n+b^n}\) is the harmonic mean of a and b, is

Concept:

The Harmonic Mean of two numbers is an average of two numbers. In particular, let a and b be two given numbers and H be the HM between them i.e., a, H, b are in HP.

\(⇒ H = \frac{2ab}{a + b}\)

Formula used:

• (x/y)ⁿ = xⁿ/yⁿ

• x⁰ = 1

Calculation:

If  \(\dfrac{a^{n+1} + b^{n+1}}{a^n+b^n}\) is the harmonic mean of a and b, then,

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

⇒ (aⁿ⁺¹ + bⁿ⁺¹)(a + b) = 2ab(aⁿ + bⁿ)

⇒ aⁿ⁺² + bⁿ⁺¹.a + an+1.b + bn+2 = 2aⁿ⁺¹b + 2abⁿ⁺¹ 

⇒ an+2 + bn+2 = an+1b + abn+1 

⇒ an+2 - an+1b  = abn+1 - bⁿ⁺² 

⇒ aⁿ⁺¹(a - b) = bⁿ⁺¹(a - b)

⇒ an+1 = bⁿ⁺¹

\(⇒ \frac{a^{n+1}}{b^{n+1}}=1\)

\(⇒\left (\frac{a}{b} \right )^{n+1}=\left (\frac{a}{b} \right )^{0}\)

⇒ n + 1 = 0

⇒ n = -1