Question

Find the mean of the first 10 even natural numbers.

Answer 1

Hint: In order to solve this problem, we need to understand that the series of first 10 even natural numbers are in arithmetic progression. We need to find the summation of the first 10 even natural numbers. The formula for that is $\text{sum}\,\left( S \right)=\dfrac{n}{2}\left( a+l \right)$, where a = first term in the series, l = last term of the series, n = number of terms. Then, the formula for finding the mean is $\text{mean = }\dfrac{\text{sum of numbers}}{\text{count of numbers}}$ .

Complete step-by-step answer:
We need to find the mean of the first 10 even natural numbers.
We will proceed by finding the sum of the first 10 even natural numbers.
Let's understand first from where the natural numbers start and where the $10^{th}$ number is.
The natural number system starts at 1. But we have asked to start from an even natural number.
Therefore, the first even natural number is 2. We know that the natural numbers arrive by one spacing between them. So, the $10^{th}$ number simply would $2\times 10=20$ .
As the spacing between the numbers is the same, we can say that the numbers are in arithmetic progression.
The formula for finding the sum of numbers in the arithmetic progression is as follows.
$\text{Sum}\,\left( S \right)=\dfrac{n}{2}\left( a+l \right).....................(i)$
Where a = first term in the series
l = last term of the series
n = number of terms
Substituting the values of a = 2, l = 20, n = 10 we get,
$S=\dfrac{10}{2}\times \left( 2+20 \right)$
Solving we get,
$\begin{align}
  & S=5\times \left( 22 \right) \\
 & =110
\end{align}$
Now the formula for mean is as follows,
$\text{Mean = }\dfrac{\text{Sum of numbers}}{\text{count of numbers}}$ ,
Substituting the values, we get,
$\text{Mean}=\dfrac{110}{10}=11$
Therefore, the mean of the first 10 even natural numbers is 11.

Note: We can solve this sum with a different approach. We can literally sum all the even numbers from 2 to 20 and then divide it by 10. We will end up with a similar answer that is 11. But in this method, there is a chance of missing a number or a calculation mistake, but in the formula, we just need to plug the values and we get the answer.