Question

The arithmetic mean of first $n$ odd natural number is:
A. ${n^2}$
B. $2n$
C. $n$
D. $3n$

Answer 1

Hint:In order to this question, to find the arithmetic mean of first $n$ odd natural numbers, we will first write the sequence of first $n$ odd natural numbers, and then we will find the sum of $n$ terms and then we will apply the formula of arithmetic mean.

Complete step by step answer:
As we know that the natural number starts from 1.
So, the first n odd natural numbers are $1,3,5,.....,n$ .
As we can see that, the above sequence of first $n$ natural numbers is an A.P.
So, the first term of an A.P, $a = 1$
Common difference of an A.P, \[d = 5 - 3 = 3 - 1 = 2\]
Now, we will find the sum of $n$ terms:-
\[{S_n} = \dfrac{n}{2}[2a + (n - 1)d] \\
\Rightarrow {S_n} = \dfrac{n}{2}[2 \times 1 + (n - 1)2] \\
\Rightarrow {S_n} = \dfrac{n}{2}[2 + 2n - 2] = {n^2} \\ \]
Now, we can apply the formula of Arithmetic Mean of $n$ terms of an A.P:-
$\text{Arithmetic Mean} = \dfrac{\text{Sum of n terms}}{\text{No. of terms}} \\
\therefore \text{Arithmetic Mean}= \dfrac{{{n^2}}}{n} = n $
Therefore, the arithmetic mean of the first $n$ odd natural number is $n$.

Hence, the correct option is C.

Note:In general, the arithmetic mean is the same as the data average. It is the group of data's representative value. If we have $n$ data points and need to compute the arithmetic mean, all we have to do is add all the numbers together and divide by the total numbers.