Question

Between $1$ and $31$, $m$ arithmetic means are inserted, so that the ratio of the $7^{th}$ and $(m-1{)}^{th}$ mean is $5:9.$ Then the value of $m$ is

• A. $12$
• B. $13$
• C. $14$
• D. $15$

Answer 1

The correct option is C $14$

Let the m means be $x_1,..,x_m$
$\therefore 1,x_1,..,x_m,31$ form an AP
$\Rightarrow 31=1+(m+2-1)d$
$\Rightarrow d=\frac{30}{m+1}...\left(1\right)$
Given $\frac{1+7d}{1+(m-1)d}=\frac{5}{9}$
$\Rightarrow 9+63d=5+5md-5d$

$\Rightarrow 4+68d=5md$

$\Rightarrow 4(1+17d)=5md$

Using $\left(1\right)$ we get,
$\Rightarrow 4\times \frac{(m+1+510)}{m+1}=\frac{150m}{m+1}$
$\Rightarrow m=14$