If $A.M.$ of two terms is $9$ and $H.M.$ is $36$,then $G.M.$ will be

For what value of $m$ is $\frac{a^{m+1}+b^{m+1}}{a^{m}+b^{m}}$ the arithmetic mean of $a$ and $b$?

Given an $A.P.$ and a $G.P.$ with positive terms, where the first and second terms of both progressions are equal. If $a_n$ and $b_n$ are the $n^{\text{th}}$ terms of the $A.P.$ and $G.P.$ respectively, then:

Find the value of $n$ so that $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ may be the geometric mean between $a$ and $b$.

If the geometric mean between $a$ and $b$ is $\frac{a^{n+1} + b^{n+1}}{a^n + b^n}$,then what is the value of $n$?

If the ratio of the Arithmetic Mean and Harmonic Mean of two positive real numbers $a$ and $b$ is $m:n$,then find the value of $a:b$.