Question

The AM of two given positive numbers is 2. If the larger number is increased by 1, the GM of number become equal to the AM of the given numbers. Then the HM of the given numbers is

• A. $\frac{3}{2}$
• B. $\frac{2}{3}$
• C. $\frac{1}{2}$
• D. none of these

Answer 1

The correct option is A $\frac{3}{2}$

Let the numbers be $a$ and $b$ where $a>b$

Given that

$a+b=4$

and $\sqrt{(a+1)b}=2$

$\Rightarrow ab+b=4=a+b$

$\Rightarrow ab=a\Rightarrow b=1(\because a\ne 0)$

and $a=3$

$\therefore HM=\frac{2\times 3\times 1}{3+1}=\frac{3}{2}$

$\therefore$ Option A is correct