Question

The arithmetic mean of first $n$ odd natural numbers is:

• A. $n^2$
• B. $2n$
• C. $n$
• D. $3n$

Answer 1

The correct option is C

$n$

Finding the mean of first $n$ odd natural numbers:

The series of first $n$ odd natural numbers will be written as:

$1,3,5,7,8,11,13.....n$

This series is in A.P.

Here, $a=1,d=(3-1)=2,$

Sum of $n$ such terms will be:

$$\begin{gathered} =\frac{n}{2}\left[2a+(n-1)d\right] \\ =\frac{n}{2}\left[2+2(n-1)\right] \\ =\frac{n}{2}\left[2n\right] \\ =n^2 \end{gathered}$$

The arithmetic mean

$$\begin{gathered} =\frac{\text{sum of}n\text{tems}}{\text{number of tems}} \\ =\frac{n^2}{n} \\ =n \end{gathered}$$

The arithmetic mean of first $n$ odd natural numbers $=n$

Hence, the correct answer is Option (A).