Question

Find the average of the first 40 natural numbers.
(a) 21
(b) 20.5
(c) 21.5
(d) None of the above.

Answer 1

Hint: We solve this problem by using the simple definition of average that is
\[Avg=\dfrac{\text{Sum of observations}}{\text{number of observations}}\]
For finding the sum of observations that is the sum of first 40 natural numbers we use the direct formula that the sum of first \['n'\] natural numbers is given as
\[\sum{n}=\dfrac{n\left( n+1 \right)}{2}\]

Complete step-by-step solution
We are asked to find the average of the first 40 natural numbers.
Let us assume that the number of observations as
\[\Rightarrow n=40\]
We know that the definition of average that is
\[Avg=\dfrac{\text{Sum of observations}}{\text{number of observations}}\]
Now, let us find the sum of observations.
We know that the direct formula that the sum of first \['n'\] natural numbers is given as
\[\sum{n}=\dfrac{n\left( n+1 \right)}{2}\]
By using the above formula the sum of first 40 natural numbers is given as
\[\begin{align}
  & \Rightarrow \sum{40}=\dfrac{40\left( 40+1 \right)}{2} \\
 & \Rightarrow \sum{40}=20\times 41 \\
 & \Rightarrow \sum{40}=820 \\
\end{align}\]
Let us assume that the average of first 40 natural numbers as \['\bar{X}'\]
Now, by using the average formula we get the average of first 40 natural numbers as
\[\Rightarrow \bar{X}=\dfrac{\sum{40}}{40}\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow \bar{X}=\dfrac{820}{40} \\
 & \Rightarrow \bar{X}=20.5 \\
\end{align}\]
Therefore, we can say that the average of first 40 natural numbers is 20.5
So, option (b) is the correct answer.

Note: We can solve this problem by using the direct formula.
The formula of the average of first \['n'\] natural numbers is given as
\[\bar{X}=\dfrac{n+1}{2}\]
We are asked to find the average of 40 natural numbers.
Let us assume that the average of first 40 natural numbers as \['\bar{X}'\]
By using the above formula we get the average of first 40 natural numbers as
\[\begin{align}
  & \Rightarrow \bar{X}=\dfrac{40+1}{2} \\
 & \Rightarrow \bar{X}=20.5 \\
\end{align}\]
Therefore, we can say that the average of first 40 natural numbers is 20.5
So, option (b) is the correct answer.