Question

If $p,q,r$ are in A.P., a is G.M. between p and q and b is G.M. between q and r, then $a^2,q^2,b^2$ are in

• A. A.P
• B. G.P.
• C. H.P
• D. none of these.

Answer 1

Answer: (A)

Since $p,q,r$ are in $A.P$.
$\therefore q=\frac{p+r}{2}...\left(1\right)$
Since $a$ is the $G.M$. between $p,q$
$\therefore b^2=qr...\left(2\right)$
Since $b$ is the $G.M$. between $q,r$
$\therefore b^2=qr...\left(3\right)$
From $\left(2\right)$ and $\left(3\right),p=\frac{a^2}{q},r=\frac{b^2}{q}$
$\therefore \left(1\right)$ gives $2q=\frac{a^2}{q}+\frac{b^2}{q}$
$\Rightarrow 2q^2=a^2+b^2$
$\Rightarrow a^2,q^2,b^2$ are in $A.P$.