Question

The mean of the first n natural number is
A. $\dfrac{n}{2}$
B. $\dfrac{{n + 1}}{2}$
C. $\dfrac{n}{2} + 1$
D. $\dfrac{{{n^2} + n + 1}}{n}$

Answer 1

Hint: According to given in the question we have to determine the mean of the first n natural number. So, first of all we have to take all the natural numbers and as we know that natural numbers are 1, 2, 3, 4, 5, …………….., n.
Now, we have to find the sum of all the n natural numbers to obtain the mean with the help of the formula to find the mean as mentioned below:

Formula used: $ \Rightarrow $Mean$ = \dfrac{{x(x + 1)}}{2}..................(A)$
Where, x is the natural numbers.
Now, we have to find the arithmetic mean of the n natural numbers which can be obtained by dividing the sum of all the numbers with the total numbers of natural numbers.

Complete step-by-step solution:
Step 1: First of all we have to take all the natural numbers and as we know that natural numbers are 1, 2, 3, 4, 5, …………….., n as mentioned in the solution hint.
Step 2: Now, we have to find the sum of all the n natural numbers to obtain the mean with the help of the formula (A) to find the mean as mentioned in the solution hint. Hence,
$ \Rightarrow n = \dfrac{{n(n + 1)}}{2}$
Step 3: Now, we have to find the arithmetic mean of the n natural numbers which can be obtained by dividing the sum of all the numbers with the total numbers of natural numbers. Hence,
$ \Rightarrow \dfrac{{\dfrac{{n(n + 1)}}{2}}}{n}$……………….(1)
Step 4: Now, to solve the expression (1) as obtained in the solution step 3 we can easily obtain the value of the mean of the first n natural number is:
$ = \dfrac{{n + 1}}{2}$
Final solution: Hence, with the help of the formula (A) as mentioned in the solution hint we have obtained the mean of the first n natural number is $ = \dfrac{{n + 1}}{2}$.

Therefore option (B) is correct.

Note: Mean can be determined by dividing the sum of all the given numbers which can be one, two or n by the total numbers.
To find the arithmetic mean of the n natural numbers which can be obtained by dividing the sum of all the numbers with the total numbers of natural numbers.