Question

If one geometric mean $G$ and two arithmetic means $p$ and $q$ be inserted between two numbers, then $G^2$ is equal to

• A. $(3p-q)(3q-p)$
• B. $(2p-q)(2q-p)$
• C. $(4p-q)(4q-p)$
• D. none of these

Answer 1

Answer: (B)

$(2p-q)(2q-p)$

Let the two numbers be $a$ and $b$, then

$G=\sqrt{ab}\Rightarrow G^2=ab$

Also, $p$ and $q$ are two $A.M.$'s between $a$ and $b$

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

$\therefore p-a=q-p$ and $q-p=b-q$

$\therefore a=2p-q$ and $b=2q-p$

$\therefore G^2=ab=(2p-q)(2q-p).$