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

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m : n - m + 1

n : n - m + 1

n - m + 1 : m

n - m + 1 : n

answer is C.

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Detailed Solution

d_{1} = \frac{2 b - a}{n + 1}, d_{2} = \frac{b - 2 a}{n + 1}

$m^{t h} A . M$ means ${(m + 1)}^{t h}$ term

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

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

$\Rightarrow n a + a + 2 b m - m a = 2 a n + 2 a + m b - 2 a m$

$\Rightarrow n a + a - m a - 2 a n - 2 a + 2 a m = m b - 2 b m$

$\Rightarrow a [n + 1 - m - 2 n - 2 + 2 m] = b (m - 2 m)$

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

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