Question

The arithmetic mean of first $n$ natural number is:

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

Answer 1

The correct option is B

$\frac{(n+1)}{2}$

Explanation for the correct option:

Finding the mean of first $n$natural numbers:

The series of first $n$ natural numbers is $1,2,3,4,5,6,.....,n$

This series is in Arithmetic progression where the consecutive terms differ by a common difference.

First term, $a=1$

Common difference, $d=1$.

$N$th term of A.P, $T_n=a+\left(n-1\right)d$

As we know, Sum of $n$ terms of A.P $=\frac{n}{2}\left(a+T_n\right)$

Therefore, the arithmetic mean of the above series will be:

$$\begin{gathered} =\frac{\text{sum of the numbers}}{\text{total number of data}} \\ =\frac{{\displaystyle \frac{n}{2}}\left(a+T_n\right)}{n} \\ =\frac{\left[{\displaystyle \frac{n}{2}}\left[2a+(n-1)d\right]\right]}{n}\left(\because T_n=a+(n-1)d\right) \\ =\frac{1}{2}\left[2+(n-1)\right] \\ =\frac{1}{2}\left[n+1\right] \end{gathered}$$

Hence, option (B) is the correct answer.