Question

Given that \(n\) number of A.Ms are inserted between two sets of numbers \(a, 2 b\) and \(2 a, b\) where \(a, b \in \mathbb{R}\). Suppose further that the \(m^{\text {th }}\) means between these sets of numbers are 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

Ans: (D)
Hint \(: 2 b=(n+2)\) th term \(=a+(n+1) d\)
$$
d=\frac{2 b-a}{n+1}
$$
$$
\therefore \text { mth mean }=a+m\left(\frac{2 b-a}{n+1}\right)
$$
Similarly \(b=(n+2)\) th term \(=2 a+(n+1) d\)
$$
d=\frac{b-2 a}{n+1}
$$
$$
\begin{aligned}
&\therefore \text { mth mean }=2 a+m\left(\frac{b-2 a}{n+1}\right) \\
&\therefore a+m\left(\frac{2 b-a}{n+1}\right)=2 a+m\left(\frac{b-2 a}{n+1}\right) \\
&\frac{a}{b}=\frac{m}{n+1-m}
\end{aligned}
$$

Answer 2

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Answer 3

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