If is the A.M. between a and b, then find the value of n.

The A.M between a and b is given by, \( \dfrac{(a + b)}{2} \)

Then according to the question,

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

2(aⁿ+bⁿ)=(a+b)\( (a^{n−1}+b^{n−1)}\)

⇒2aⁿ+2bⁿ = \(a (a^{n−1}+b^{n−1})+b (a^{n−1}+b^{n−1})\)

⇒\(2a^n+2b^n = a^n+ab^{n−1}+ba^{n−1}+b^n\)

⇒\( 2a^n+2b^n−a^n−ab^{n−1}−ba^{n−1}−b^n=0\)

⇒\( a^n−ba^{n−1}+b^n−ab^{n−1}=0\)

⇒\( a^{n−1}(a−b)−b^{n−1}(a−b)=0\)

\( a^{n−1}−b^{n−1}=0\; or\; a−b=0\)

\( a^{n−1}=b^{n−1} \;or\; a=b\; but\; a\neq b\)

\( \dfrac{a^{n-1}}{b^{n-1}}=1\)

\( ( \dfrac{a}{b})^{n-1}= ( \dfrac{a}{b})^0\)

Thus, the value of n is 1.