What is the mean of first n odd natural numbers

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7. Find the mean of first n odd natural numbers​

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Mean of first n odd numbers is n.

  • Odd number starts from 1 .
  • The series of odd numbers is 1 , 3 , 5 , 7, 9 , and so on.
  • We have to find the mean of first n odd numbers.
  • To obtain the mean of n numbers, we find the sum of n numbers and then divide the sum by n.
  • Now to find the sum of n odd numbers , we see that the given odd numbers are in AP with first term(a) = 1 and common difference(d) = 2.
  • Applying the formula of sum of n terms of AP,

Sum of n terms of AP = \frac{n}{2}[2a + (n-1)d]

S_n = \frac{n}{2}[2a + (n-1)d]

  • Substituting the value of a and d in the equation, we get

S_n = \frac{n}{2}[2(1) + (n-1)(2)]

S_n = \frac{n}{2}[2 + 2n - 2]

S_n = \frac{n}{2}[2n]

S_n = n^2

  • Now calculating mean of n odd number,

Dividing the sum by n, we get

Mean of n odd numbers = \frac{S_n}{n} = \frac{n^2}{n} = n.


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