Question

If one $G.M.$ G and two geometric means $p$ and $q$ be inserted between any two given numbers, then $G^2=(2p-q)(2q-p)$.

• A. True
• B. False

Answer 1

The correct option is A True

Let the given two numbers be $a,b$

$G^2=pq$ $.............\left(1\right)$

Since,

$a,p,q,b$ are in $A.P$

$b=a+3d$

$\Rightarrow d=\frac{b-a}{3}$

Now,

$p=a+d=a+\frac{b-a}{3}=\frac{2a+b}{3}$

$q=p+d=\frac{2a+b}{3}+\frac{b-a}{3}=\frac{a+2b}{3}$

Now,

$2p-q=\frac{3a}{3}=a$ $.............\left(2\right)$

$p-2q=-b$ $...............\left(3\right)$

Therefore,

$(2p-q)(p-2q)=-ab$

$(2p-q)(p-2q)=-G^2$

$G^2=(2p-q)(2q-p)$

Hence, this is the answer.