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

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$n - m + 1 : m$

$n - m + 1 : n$

$m : n - m + 1$

$n : n - m + 1$

answer is C.

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Detailed Solution

Let $A_{1}, A_{2} ..... A_{n}$ be arithmetic means between a and 2b , then $A_{m} = a + m (\frac{2 b - a}{n + 1})$
Again , let $B_{1}, B_{2} ........ B_{n}$ be arithmetic means
Between 2a and b then $B_{m} = 2 a + m (\frac{b - 2 a}{n + 1})$
Now , $A_{m} = B_{m} \Rightarrow a + m (\frac{2 b - a}{n + 1}) = 2 a +$

$m (\frac{b - 2 a}{n + 1}) \Rightarrow m (\frac{b + a}{n + 1}) = a \Rightarrow \frac{a}{b} = \frac{m}{n - m + 1}$