Question

If $A_1$ be the A.M. and $G_1,G_2$ be two $G$ Ms between two positive numbers $a$ and $b$, then $\frac{G_1^3+G_2^3}{G_1{G}_2{A}_1}$ is equal to

• A. $4$
• B. $1$
• C. $2$
• D. None of these

Answer 1

The correct option is C $2$

Since $A_1$ is the arithmetic mean between a and b

$\therefore A_1=\frac{a+b}{2}$

Also $G_1,G_2$ are the geometric means between a and b

so, a,$G_1,G_2,b$ are in GP

n=4,first term=a,last term=b

$b=a(r{)}^{4-1}=a{r}^3⟹r=(\frac{b}{a}{)}^{\frac{1}{3}}$

$\therefore G_1=ar=a(\frac{b}{a}{)}^{\frac{1}{3}}$

$G_2=a{r}^2=a(\frac{b}{a}{)}^{\frac{2}{3}}$

$\therefore \frac{G_1^3+G_2^3}{G_1{G}_2{A}_1}=\frac{a^3\frac{b}{a}+a^3(\frac{b}{a}{)}^2}{a^2\left(\frac{b}{a}{)}^1\right(\frac{a+b}{2})}$

$=\frac{a^3\frac{b}{a}(1+\frac{b}{a}}{a^2\frac{b}{a}\frac{a+b}{2}}$

$=2\frac{a+b}{a+b}=2$