Question

If one G.M. '$G$' and two arithmetic means $p$ and $q$ be inserted between any two given number then $G^2=$

• A. $(2p-q)(2q-p)$
• B. $(2p-q)(q-2p)$
• C. $\frac{2p+q}{2q+p}$
• D. $\frac{2p+q}{q+2p}$

Answer 1

The correct option is A $(2p-q)(2q-p)$

$a,G,b$ $a,p,q,b$

$a=\sqrt{ab}$ $p=\frac{2ab}{3},q=\frac{2b+a}{3}$

$G^2=ab$

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

$2a+b=3p$

$a+2b=3q\times 2$

$b=2q-p$

$2a+b=3p$ $a=3q-2(2q-p)$

$2a+4b=6q$ $=3q-4q+2p$

$---$ $a=2p-q$

_____________

$-3b{3}^2(p-2q)$