Question

Let $p,p_1$ be A.M. and G.M. between $a$ and $b$ respectively and $q,q_1$ be the A.M. and G.M. between $b$ and $c$ respectively where $a,b,c>0$. If $a,b,c$ are in A.P., then which of the following is CORRECT?

• A. $p^2-q^2=q_1^2-p_1^2$
• B. $p^2-q^2=p_1^2-q_1^2$
• C. $p^2+q^2=p_1^2-q_1^2$
• D. $p^2-q^2=p_1^2+q_1^2$

Answer 1

The correct option is B $p^2-q^2=p_1^2-q_1^2$

Given :
$p=\frac{a+b}{2},q=\frac{b+c}{2}{p}_1=\sqrt{ab},q_1=\sqrt{bc}$

$a,b,c$ are in A.P.
$\Rightarrow 2b=a+c$
Now, $p^2-q^2=\frac{1}{4}(a^2+b^2+2ab-b^2-c^2-2bc)\Rightarrow p^2-q^2=\frac{1}{4}(a^2-c^2+2b(a-c\left)\right)\Rightarrow p^2-q^2=\frac{(a-c)(a+c+2b)}{4}\Rightarrow p^2-q^2=\frac{(a-c)(2b+2b)}{4}\Rightarrow p^2-q^2=b(a-c)\Rightarrow p^2-q^2=ab-bc\therefore p^2-q^2=p_1^2-q_1^2$