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Q.
If one geometric mean $G$ and two arithmetic means $p$ and $q$ be inserted between two numbers, then $G^{2}$ is equal to
Sequences and Series
Solution:
Let the two numbers be $a$ and $b$, then
$G=\sqrt{a b} $ or $ G ^{2}=a b$
Also, $p$ and $q$ are two A.M.s between $a$ and $b$.
$\therefore a, p, q, b$ are in A.P.
$\therefore p-a=q-p$ and $q-p=b-q$
$\therefore a=2 p-q$ and $b=2 q-p$
$\therefore G^{2}=a b=(2 p-q)(2 q-p) .$