Question

The arithmetic mean of the first n natural numbers is
$$\begin{gathered} A){\displaystyle \frac{\left(n+1\right)}{2}} \\ B){\displaystyle \frac{\left(n-1\right)}{2}} \\ C){\displaystyle \frac{n}{2}} \\ D)\text{none of the above} \end{gathered}$$

Answer 1

Hint: To proceed with a solution we need the sum of n natural numbers which will be helpful to solve the arithmetic mean of the first n natural numbers where the formula of arithmetic mean of the first n natural numbers includes the sum of n natural numbers.

We know that
First n natural numbers are 1, 2, 3, 4……………………..n
We also know that sum of n natural numbers =${\displaystyle \frac{n\left(n+1\right)}{2}}$
Hence we know that
Arithmetic mean of n natural numbers = ${\displaystyle \frac{\text{Sum of natural numbers}}{\text{Total natural numbers}}}$
Arithmetic mean = ${\displaystyle \frac{{\displaystyle \frac{n\left(n+1\right)}{2}}}{n}}$
Arithmetic mean = ${\displaystyle \frac{n+1}{2}}$
Therefore arithmetic mean is ${\displaystyle \frac{n+1}{2}}$
 Option A is the correct answer.

NOTE: In this problem to get the arithmetic mean of n natural numbers we need to know the sum of n natural numbers where the formula of arithmetic mean includes .After getting the values we have substituted in formula and proceeded on calculation part. Here the calculation is a simple cancellation of n terms where we get the answer as ${\displaystyle \frac{n+1}{2}}$.