Question

The sum of $n$ arithmetic means between$a$ and $b$ is

• A. $\frac{n(a+b)}{2}$
• B. $n(a+b)$
• C. $\frac{(n+1)(a+b)}{2}$
• D. $(n+1)(a+b)$

Answer 1

The correct option is A

$\frac{n(a+b)}{2}$

Explanation for the correct option:

Find the required sum.

We have $a$ and $b$two given numbers.

Let,

$A_1,A_2,A_3.............A_n$​ be Arithmetic means between $a$ and $b$.

So, $a,A_1,A_2,A_3.............A_n,b$ are in A.P.

$a+A_1+A_2+A_3+......A_n+b=\frac{(n+2)}{2}(a+b)$ (first term is $=a$, last term$=b$, number of term$=n+2$)

$$\begin{gathered} \Rightarrow A_1+A_2+A_3+......A_n=\frac{(n+2)(a+b)}{2}-(a+b) \\ =(a+b)\left(\frac{n+2}{2}-1\right) \\ =(a+b)\left(\frac{n+2-2}{2}\right) \\ =\frac{n\left(a+b\right)}{2} \end{gathered}$$

Therefore, the sum of $n$ arithmetic means between$a$ and $b$ is $\frac{n\left(a+b\right)}{2}$.

Hence, option (A) is the correct answer.