Question

If $g_1, g_2$ are two geometric means and $a_1$ is the arithmetic mean between two positive numbers, then the value of $\frac{g_{1}^{2}}{g_{2}} + \frac{g_{2}^{2}}{g_{1}}$ is

• A. $2a_1$
• B. $a_1$
• C. $\frac{a_{1}}{2}$
• D. $3a_1$

Answer 1

Answer: (A)

$2a_1$

Answer 2

Let $a, g_{1}, g_{2}, b$ are in $G. P$. $\therefore g_{1}^{2} = ag_{2}$ $\Rightarrow \frac{g_{1}^{2}}{g_{2}} = a \quad ...\left(i\right)$ Also, $g_{2}^{2} = bg_{1}$ $\Rightarrow \frac{g_{2}^{2}}{g_{1}} = b \quad...\left(ii\right)$ Adding $\left(i\right)$ and $\left(ii\right)$, we get $\frac{g_{1}^{2}}{g_{2} }+\frac{g_{2}^{2}}{g_{1}} = a+b$ $= 2a_{1}$ [ $\because a, a_{1}, b$ are in $A.P$.]