Question

If $5$ geometric means are inserted between $3\frac{5}{9}$ and $40\frac{1}{2}$ then the ratio of greatest to least mean is.

• A. $1$
• B. $\left(\frac{3}{2}\right)$
• C. ${\left(\frac{3}{2}\right)}^2$
• D. ${\left(\frac{3}{2}\right)}^4$

Answer 1

The correct option is D ${\left(\frac{3}{2}\right)}^4$

Let $G_1,G_2,G_3,G_4\&G_5$ are the five geometric means inserted between $3\frac{5}{9}$ and $40\frac{1}{2}$

then $\frac{32}{9},G_1,G_2,G_3,G_4,G_5,\frac{81}{2}$ is a G.P with $a=\frac{32}{9}$ and $a_7=\frac{81}{2}$

Now ,

$a_7=\frac{81}{2}$

$\Rightarrow a{r}^6=\frac{81}{2}$

$\Rightarrow \frac{32}{9}{r}^6=\frac{81}{2}$

$\Rightarrow r^6=\frac{81}{2}\times \frac{9}{32}$

$\Rightarrow r^6={\left(\frac{3}{2}\right)}^6$

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

So, $G_1$ and $G_5$ are the least and greatest mean respectively

therefore ,

$\frac{G_5}{G_1}=\frac{a_6}{a_2}=\frac{a{r}^5}{ar}=r^4$

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