Question

$n$ arithmetic means are inserted in between $x$ and $2y$ and then between $2x$ and $y.$ In case the $r^{th}$ means in each case be equal, then find the ratio $\frac{x}{y}.$

• A. $\frac{r}{n+r+1}$
• B. $\frac{n}{n+r+1}$
• C. $\frac{n}{n-r+1}$
• D. $\frac{r}{n-r+1}$

Answer 1

The correct option is D $\frac{r}{n-r+1}$

Let $n$ AMs are inserted between $a$ and $b$

i.e $a,x_1,x_2,...x_n,b$

$b=a+(n+1)d$

$\therefore \frac{b-a}{n+1}=d$

$\therefore x_r=T_{r+1}=a+rd=a+\frac{r(b-a)}{n+1}=\frac{a(n-r+1)+rb}{n+1}...\left(1\right)$

Put $a=x$ and $b=2y$ and then again put $a=2x$ and $b=y$ and equate the results as the two means are equal

$\therefore \frac{x(n-r+1)+2yr}{n+1}=\frac{2x(n-r+1)+yr}{n+1}$

$\therefore x(n-r+1)=yr$

$\Rightarrow \frac{x}{y}=\frac{r}{n-r+1}$

Hence, option D.