Question

What is the mean of first n odd natural numbers?
$
  {\text{A}}{\text{. n}} \\
  {\text{B}}{\text{. }}\dfrac{{{\text{n + 1}}}}{2} \\
  {\text{C}}{\text{. }}\dfrac{{{\text{n}}\left( {{\text{n + 1}}} \right)}}{2} \\
  {\text{D}}{\text{. n + 1}} \\
$

Answer 1

Hint – To find the mean, we observe the numbers are in arithmetic progression. Then we find the ${{\text{n}}^{{\text{th}}}}$ term of the progression and then we find the sum of first n odd natural numbers by using the first and the ${{\text{n}}^{{\text{th}}}}$ odd number.

Complete step-by-step answer:

We know the first n odd terms are in arithmetic progression with common difference (d) = 2, i.e. the difference between any two consecutive numbers is two.

We know, the first odd natural number (a) = 1
We know the ${{\text{n}}^{{\text{th}}}}$ odd natural number can be given as (${{\text{a}}_{\text{n}}}$) = a + (n-1) d
Where a is the first term, n is the number of terms of the progression and d is the common difference.
= 1 + (n – 1) x 2
= 1 + 2n -1
= 2n -1
Where n belongs to the set of natural numbers = (1, 2, 3…)

We know, sum of first n terms in AP can be given as = $\dfrac{{\text{n}}}{2}\left( {{\text{a + }}{{\text{a}}_{\text{n}}}} \right)$
Therefore, sum of first n odd natural numbers
= $\dfrac{{\text{n}}}{2}\left( {1{\text{ + 2n - 1}}} \right)$
= $\dfrac{{\text{n}}}{2}\left( {{\text{2n}}} \right)$
= ${{\text{n}}^2}$

We know Mean can be given as = $\dfrac{{{\text{Sum of all terms}}}}{{{\text{Total number of terms}}}}$

Mean of first n odd natural numbers = $\dfrac{{{\text{Sum of first n odd natural numbers}}}}{{{\text{Total number of first n odd natural numbers}}}}$

Mean of first n odd natural numbers = $\dfrac{{{{\text{n}}^2}}}{{\text{n}}}$
⟹Mean of first n odd natural numbers = n

Hence, Mean of first n odd natural numbers is n. Option A is the correct answer.

Note – The key in solving such types of problems is to identify that the numbers are in arithmetic progression. Then the next crucial steps are finding the ${{\text{n}}^{{\text{th}}}}$ term and sum of terms using the formulas of AP. The sum of first n odd natural numbers divided by the first n odd natural numbers gives the mean.