Question

Find the mean of first n natural numbers.

Answer 1

Hint:Mean can be calculated by dividing sum of numbers (observation) by number of observations.
For example, if your class has 3 students and they scored 60, 70, 80 marks respectively.
Average marks that your class has scored is:
Mean = \[\dfrac{{{\text{Sum of all numbers}}}}{{{\text{No}}{\text{. of observation}}}}\]
             = $\dfrac{{60 + 70 + 80}}{3}$
             = $\dfrac{{210}}{3}$
             = 70

Complete step-by-step answer:

Now coming to the exact question, here we have two calculate the mean of first n natural numbers.
If we want to calculate mean of first n natural numbers:
First n natural numbers = 1, 2, 3, 4,5 ……………. n
Sum of n natural numbers = (1+2+3+……+n) (Eq 1)
                                                 = $\dfrac{{{\text{n}}({\text{n}} + 1)}}{2}$ (Eq 2)
Total no. of observations = n
Now, we know that
Mean = \[\dfrac{{{\text{Sum of all numbers}}}}{{{\text{No}}{\text{. of observation}}}}\].
Putting the values from equation 1 and 2, we get:
        Mean =$\dfrac{{1 + 2 + 3 + ..... + {\text{n}}}}{{\text{n}}}$
                      = $\dfrac{{{\text{n}}({\text{n}} + 1)}}{{2 \times {\text{n}}}}$
                       = $\dfrac{{(n + 1)}}{2}$


Therefore, we can write:
Mean of first n natural numbers = $\dfrac{{(n + 1)}}{2}$

Note: In this type of question, first, we should remember the formula to find the sum of n natural numbers which is given as ; S= $\dfrac{{n(n + 1)}}{2}$. After this to find the mean, you have to divide the sum by the total number of entities, which is n in this case.