Question

If $n$ arithmetic means are inserted between $20$ and $80$ such that the ratio of the first mean to the last mean is $1:3,$ then the value of $n$ is

• A. $12$
• B. $6$
• C. $11$
• D. $5$

Answer 1

The correct option is C $11$

Let $A_1,A_2,\dots ,A_n$ be $n$ A.M.s between $20$ and $80$.
Given, $A_1:A_n=1:3$
$\Rightarrow 3A_1=A_n\Rightarrow 3[20+\frac{1(80-20)}{n+1}]=20+\frac{n(80-20)}{n+1}\Rightarrow 60+\frac{180}{n+1}=20+\frac{60n}{n+1}\Rightarrow \frac{60(n-3)}{n+1}=40\Rightarrow 20n=220\therefore n=11$