Question

If three numbers are in GP then the numbers obtained by adding the middle number to each of the three numbers are

• A. AP
• B. GP
• C. HP
• D. none of these

Answer 1

The correct option is C HP

Let the numbers be $a,b,c$

$\therefore b^2=ac$

Numbers obtained by adding middle number to each number are $a+b,2b,b+c$
Let us consider $\frac{1}{a+b}+\frac{1}{c+b}$

$=\frac{a+c+2b}{ac+b^2+bc+ac}$

$=\frac{a+c+2b}{2b^2+ab+bc}$

$=\frac{a+c+2b}{\left(b\right)(a+c+2b)}$

$=\frac{1}{2b}$

$\therefore a+b,2b,b+c$ are in HP