Question

If one $A.M.A$ and two geometric means $p$ and $q$ be inserted between any two given numbers, then show that $p^3+q^3=2Apq$.

Answer 1

Let the numbers be $x$ and $y$
$A$ is their $A.M.$ $2A=x+y$
Also $p,q$ are two geometric means between $x$ and $y$
$\therefore$ $x,p,q,y$ are in $G.P.$
Since $x,p,q$ are in $G.P.$ $\therefore p^2=xq$ or $p^3=xpq$
Since $p,q,y$ are in $G.P.$ $\therefore q^2=py$ or $q^3=ypq$
Adding, $p^3+q^3=pq(x+y)=2Apq$, by $\left(1\right)$
$\therefore p^3+q^3=2Apq$.