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 $x/y.$

• A. $\frac{n}{n-r+1}$
• B. $\frac{r}{n+r+1}$
• 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 $n$ means are inserted between $a$ and $b$
$a,,x_1,x_2,...x_n,b$
Then last term $b=t_{n+2}=a+(n+1)d$
$\Rightarrow \frac{b-a}{n+1}=d$
And general term
$\Rightarrow x_r=T_{r+1}=a+rd=a+\frac{r(b-a)}{n+1}=\frac{a(n-r+1)+rb}{n+1}$
Now put $a=x,b=2y$ and $a=2x,b=y$ , and equate their means
$\frac{x(n-r+1)+2yr}{n+1}=\frac{2x(n-r+1)+yr}{n+1}$

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