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.

Explanation

GM=`sqrt(ab)`

`=>(​ab)=(a^(n+1)+b^(n+1))/(​a^n+b^n)`

`=>(ab)^(1/2)[a^n+b^n]=a^(n+1)+b^(n+1)`

`=>a^(1/2​+n)b^(1/2​)+a^(1/2)​b^(1/2​+n)=a^(n+1)+b^(n+1)`

`=>a^(1/2​+n)[b^(1/2)​−a^(1/2)​]=b^(n+1/2​)[b^(1/2)−a^(1/2)​]`

`=>a^(1/2​+n)=(b^(n+1/2​)[b^(1/2)​−a^(1/2)​]​)/((b^(1/2)​−a^(1/2)​)`

`=>a^(1/2​+n)=b^(n+1/2​)`

`=>(a/b​)^(1/2​+n)=1`

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