The value of $n$ for which $\dfrac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ is the geometric mean of $a$ and $b$ is:

Setting the expression equal to √(ab) and simplifying gives a^{n+1}+b^{n+1} = √(ab)(a^n+b^n); this holds for n = −1/2.