Question

If one geometric mean G and two arithemtic 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. $(4p+q)(4q+p)$

Answer 1

Answer: (B)

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

Let $a$ and $b$ be two numbers.
Since $G$ is G.M. between $a$ and $b$, implies $G^2=ab$.
Also, $p$ and $q$ are A.M's between $a$ and $b$ implies $a,p,q,b$ are in A.P.
$\Rightarrow 2p=a+q$ and $2q=p+b$
$\Rightarrow a=2p-q$ and $b=2q-p$
$\Rightarrow G^2=(2p-q)(2q-p)$
So, option B is correct.