Question

If $3$ geometric means are inserted between $2\frac{1}{4}$ and $\frac{4}{9}$ then the ratio of greatest to least mean is

• A. $1$
• B. ${\left(\frac{3}{2}\right)}^2$
• C. $\left(\frac{3}{2}\right)$
• D. $Noneofthese$

Answer 1

The correct option is B ${\left(\frac{3}{2}\right)}^2$

Let $G_1,G_2,G_3$ be three geometric mean inserted between $2\frac{1}{4}\&\frac{4}{9}$

then

$\frac{9}{4},G_1,G_2,G_3,\frac{4}{9}$ is a G.P with $a=\frac{9}{4}\&a_5=\frac{4}{9}$

$a_5=\frac{4}{9}$

$\Rightarrow a{r}^4=\frac{4}{9}$

$\Rightarrow \frac{9}{4}{r}^4=\frac{4}{9}$

$\Rightarrow r^4={\left(\frac{4}{9}\right)}^2$

$\Rightarrow r^4={\left(\frac{2}{3}\right)}^4$

$\Rightarrow r=\frac{2}{3}<1$

So, $G_1$ & $G_3$ are the greatest and least mean respectively

Now , $\frac{G_1}{G_3}=\frac{a_2}{a_4}=\frac{ar}{a{r}^3}$

$=\frac{1}{r^2}$

$={\left(\frac{3}{2}\right)}^2$