Question

If three numbers are in HP then the numbers obtained by subtracting half of the middle number from each of them are in

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

Answer 1

The correct option is B GP

Let the numbers be $a,b,c$

Given $b=\frac{2ac}{a+c}$ .......(1)
Numbers obtained by subtracting $\frac{b}{2}$ from each are

$a-\frac{b}{2},\frac{b}{2},c-\frac{b}{2}$

Now, $(a-\frac{b}{2})(c-\frac{b}{2})=ac-(c+a)\frac{b}{2}+\frac{b^2}{4}$

$=ac-ac+\frac{b^2}{4}$ from (1)

$\Rightarrow \frac{b^2}{4}=(a-\frac{b}{2})(c-\frac{b}{2})$

$\therefore$ The new numbers are in GP