Find the value of n so that `(a^(n+1)+b^(n+1))/(a^n+b^n)`may be the geometric mean between a and b.

Options

A. `n=(1)/(2)`
B. `n=-(1)/(3)`
C. `n=-(1)/(2)`
D. `n=(1)/(3)`

Answer

C

Explanation

`(a^(n+1)+b^(n+1))/(a^(n)+b^(n))`= GM between a and b
`implies [ (a^(n+1)+b^(n+1))/(a^(n)+b^(n))]=sqrt(ab)`
`a^(n+1)+b^(n+1)=(a^(n)+b^(n))a^((1)/(2))b^((1)/(2))`
`impliesa^(n+1)+b^(n+1)=a^((n+(1)/(2)))b^((1)/(2))=a^(1/(2))b^((n+(1)/(2)))-b^(n+1)`
`impliesa^((n+(1)/(2)))[a^((1)/(2))-b^((1)/(2))]=n^((n+(1)/(2)))[a^((1)/(2))-b^((1)/(2))]`
`implies a^((n+(1)/(2)))=b^((n+(1)/(2)))`
`implies n+(1)/(2)=0implies n=-(1)/(2)`