Question

The sum of first $10$ natural numbers is__________ .
A. $100$
B. $55$
C. $50$
D. $90$

Answer 1

Hint: To solve this question we need to know the concept of Arithmetic Progression. A series is said to be in A.P when the common difference between all the two terms are the same. We find the value of common difference by subtracting the two consecutive terms in the series. To find the sum of the numbers in the A.P series the formula we use is, $\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$.

Complete step by step solution:
To solve this question we need to know about the natural number, as per the question we need to find the sum of natural numbers till the $10th$ term. We know that the natural number starts from $1$, so we need to find the sum till $10$. The natural numbers series is in Arithmetic Progression, AP as the common difference of the terms is $1$. Common difference is found as ${{\left( n+1 \right)}^{th}}-{{\left( n \right)}^{th}}$ . We may check by subtracting $9th$ from $10th$ term which is
$\Rightarrow 10-9$
$\Rightarrow 1$
So the common difference is $1$.
We can find the sum by using the formula $\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$ , here $''a''$ is the first term, $''n''$ is the number of terms and $''d''$ is the common difference of the AP. The values of these terms given in the question are$a=1,n=10,d=1$. Substituting these values in the same formula, we get:
$\Rightarrow \dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$
$\Rightarrow \dfrac{10}{2}\left( 2\times 1+\left( 10-1 \right)1 \right)$
Now using BODMAS to calculate the above expression we get:
$\Rightarrow 5\left( 2+\left( 9 \right)1 \right)$
$\Rightarrow 5\left( 2+9 \right)$
Adding the number inside the bracket and then multiplying it with $5$ , we get:
$\Rightarrow 5(11)$
Bracket here means the number $5$ is getting multiplied with $11$ , so the answer comes out to be
$\Rightarrow 55$
$\therefore $ The sum of first $10$ natural numbers is $55$, which means option $B$ is correct.

So, the correct answer is “Option B”.

Note: When numbers are in a certain series the calculation of the sum of the numbers becomes easier. So the first step for the calculation of these types of the series, try finding the relation between each term. If we are able to find a certain relation between the terms in the series formula can directly be applied in that situation.