For what value of `n, (a^(n+1)+b^(n+1)/(a^n+b^n), a!=b` is the A.M. of a and b.

Explanation

A.M. of a and b = `(a + b)/(2)`
Given , `(a^(n+1) + b^(n+1))/(a^(n) + b^(n) ) = (a + b)/(2) `
`rArr 2a ^(n+1) + 2b ^(n+1) = a^(n+ 1) + b^(n+1) + a^(n) b + ab^(n) `
`rArr a^(n+1) + b^(n + 1) - ab^(n) - a^(n) b = 0 `
`rArr a^(n) (a - b) b^(n) (a - b) = 0 `
`rArr (a - b)(a^(n) - b^(n)) = 0 `
Since ` a ne b , therefore a - b ne 0 `
` therefore a^(n) - b^(n) = 0 `
` rArr a^(n) = b^(n)`
`rArr ((a)/(b))^(n) = 1 (because ((a)/(b))^(0) = 1)`
` therefore n = 0 `