Question

Given that $n A.M.'s$ are inserted between two sets of numbers $a, 2b$ and $2 a, b$, where $a, b \in R$. Suppose further that $mth$ mean between these sets of numbers is same, then the ratio, $a : b$ equals

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

Answer 1

Answer: (D)

$m : n - m + 1$.

Answer 2

Since $n \,A.M.??$ are insetted between $a$ and $26$, we have $mth$ mean $= a+\frac{m\left(2b-a\right)}{n+1}$ Similarly, $mth$ mean between $2\, a$ and $b$ $= 2a+\frac{m\left(b-2a\right)}{n+1}$ By the given condition $a+\frac{m\left(2b-a\right)}{n+1} = 2a + \frac{m\left(b-2a\right)}{n+1}$ $\Rightarrow a= \frac{m}{n+1} \left(2b -a -b +2a\right) = \frac{m}{n+1}\left(a+b\right)$ $\Rightarrow a\left(1- \frac{m}{n+1}\right) = \frac{mb}{n+1}$ $\Rightarrow \frac{a\left(n+1-m\right)}{n+1} = \frac{mb}{n+1}$ $\Rightarrow \frac{a}{b} = \frac{m}{n-m+1}$