Question:
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
(a) (2p − q) (p − 2q)
(b) (2p − q) (2q − p)
(c) (2p − q) (p + 2q)
(d) none of these
Solution:
(a) $(2 p-q)(p-2 q)$
Let the two numbers be $a$ and $b$.
$a, p, q$ and $b$ are in A.P.
$\therefore p-a=q-p=b-q$
$\Rightarrow p-a=q-p$ and $q-p=b-q$
$\Rightarrow a=2 p-q$ and $b=2 q-p$ ...(i)
Also, $a, G$ and $b$ are in G.P.
$\therefore G^{2}=a b$
$\Rightarrow G^{2}=(2 p-q)(2 q-p)$