Question

If n arithmetic means are inserted between two sets of numbers a, 2b and 2a, b, where $a,bϵR.$ Suppose that $m^{th}$ arithmetic mean between these two sets of numbers is same, then the ratio a:b equals

• A. n - m + 1:m
• B. m:n - m + 1
• C. n:n - m + 1
• D. n - m + 1:n

Answer 1

The correct option is B m:n - m + 1

Let $A_1,A_2.....A_n$ be airthmetric means between a and 2b , then $A_m=a+m\left(\frac{2b-a}{n+1}\right)$
Again, let $B_1,B_2.....B_n$ be arithmetic means
Between 2a and b then $B_m=2a+m\left(\frac{b-2a}{n+1}\right)$
Now, $A_m=B_m\Rightarrow a+m\left(\frac{2b-a}{n+1}\right)=2a+$
$m\left(\frac{b-2a}{n+1}\right)\Rightarrow m\left(\frac{b+a}{n+1}\right)=a\Rightarrow \frac{a}{b}=\frac{m}{n-m+1}$