Question

Calculate the mean of first $n$ natural numbers ?

Answer 1

Step 1 : Finding the sum of $\mathbf{}\mathbf{n}\mathbf{}$ natural numbers

The sum of the $\mathrm{n}$natural numbers is

$$\begin{gathered} =1+2+3+4+.....+\mathrm{n} \\ =\frac{\mathrm{n}\left(\mathrm{n}+1\right)}{2} \end{gathered}$$

Step 2: Find the mean of the first $\mathbf{}\mathbf{n}\mathbf{}$natural numbers

We know that the mean of an observation $=\frac{\mathrm{Sum}\mathrm{of}\mathrm{all}\mathrm{observation}}{\mathrm{Number}\mathrm{of}\mathrm{observation}}$

So, the mean of the first $\mathbf{}\mathbf{n}\mathbf{}$natural numbers $=\frac{\mathrm{n}\left(\mathrm{n}+1\right)}{2\mathrm{n}}$

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

Hence, the mean of the first $\mathbf{}\mathbf{n}\mathbf{}$natural numbers is $\frac{(\mathrm{n}+1)}{2}$