Question

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

• A. $14$
• B. $10$
• C. $7$
• D. $3$

Answer 1

Answer: (A)

$14$

Let the m means be $x_1,..,x_m$

$\therefore 1,x_1,..,x_m,31$ form an AP.

$\Rightarrow 31=1+(m+2-1)d$

$\Rightarrow d=\frac{30}{m+1}$

Given $\frac{1+7d}{1+(m-1)d}=\frac{5}{9}$

$\Rightarrow 9+63d=5+5md-5d$

$\Rightarrow 4\frac{(m+1+510)}{m+1}=\frac{150m}{m+1}$

$\Rightarrow m=14$