Question

Find the arithmetic mean of the first 10 odd natural numbers.

Answer 1

Hint: In order to solve this question, we will first categorize the first 20 natural numbers, then opt for 10 natural odd numbers from them. And finally, calculate the arithmetic mean by using the below stated formula.
\[\text{Arithmetic Mean}=\dfrac{\text{Sum of the first 10 odd numbers}}{10}\]

Complete step by step answer:
We know that the natural numbers are the numbers which start with 1. Basically, they are 1, 2, 3, 4, 5, ….. Let us first write the first 20 natural numbers, they are as below.
\[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20\]
Let us select the odd numbers from them. The odd numbers are the numbers which are not divisible by 2. Now, selecting the odd numbers from the above list, we have the numbers as
\[1,3,5,7,9,11,13,15,17,19\]
The Arithmetic mean of 100 odd natural numbers is given by the formula
\[\text{Arithmetic Mean}=\dfrac{\text{Sum of the first 10 odd numbers}}{10}\]
\[\Rightarrow \text{Arithmetic Mean}=\dfrac{1+3+5+7+9+11+13+15+17+19}{10}\]
\[\Rightarrow \text{Arithmetic Mean}=\dfrac{100}{10}\]
\[\Rightarrow \text{Arithmetic Mean}=10\]

Therefore, the arithmetic mean of the first 10 odd natural numbers is given by 10.

Note: A possibility of the mistake in this question can be directly adding all the numbers from 1 to 20 and then dividing it by ‘2’ to get the sum of the odd numbers till 20. This is wrong as the sum of the n terms is given by \[\dfrac{n\left( n+1 \right)}{2},\] then the sum of the 20 terms will be \[\dfrac{20\left( 21 \right)}{2}=10\left( 21 \right)=210.\] Now dividing by 2 gives, \[\dfrac{210}{2}=105\ne 100.\] We had the sum of the first 10 odd numbers as 100 and not 105. So, this method is wrong.