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Q.
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
Sequences and Series
Solution:
Since $p,q,r$ are in $A.P$.
$ \therefore q=\frac{p+r}{2} \quad...\left(1\right)$
Since $a$ is the $G.M$. between $p, q$
$ \therefore b^{2}=qr\quad ...\left(2\right) $
Since $b$ is the $G.M$. between $q,r$
$ \therefore b^{2} = qr \quad...\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 $A.P$.