Question

If $A$ is the single A.M. between two numbers $a$ and $b$ and $S$ is the sum of $n$ A.M.'s between them, then $\frac{S}{A}$ depends upon

• A. $n,a,b$
• B. $n,a$
• C. $n,b$
• D. $n$

Answer 1

The correct option is D $n$

A is the single A.M. between numbers $a$ and $b$.

So, $A=\frac{a+b}{2}$

If there are $n$ A.M.'s between $a$ and $b$, they would be

$a+\frac{b-a}{n+1},a+\frac{2(b-a)}{n+1},...,a+\frac{(n-1)(b-a)}{n+1},a+\frac{n(b-a)}{n+1}$ excluding $a$ and $b$.

The sum of these would be

$na+\frac{n(n+1)(b-a)}{2(n+1)}=\frac{2na+nb-na}{2}=\frac{n(a+b)}{2}$

$\therefore \frac{S}{A}=n$