# If p, q be two A.M.'s and G be one G.M. between two numbers,

Question:

If pq 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.

$\therefore G^{2}=a b$

$\Rightarrow G^{2}=(2 p-q)(2 q-p)$