Question

What is the sum of the first 10 positive integers?

Answer 1

Hint: To find the sum of the first 10 positive integers we need to find what the first 10 positive integers are. Positive means there won’t be any negative and 0 term. The first 10 positive integers are written as follows: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Now, we are going to add these 10 positive integers using the summation formula of A.P. to get the sum of 10 positive integers.

Complete step-by-step solution:

The series of first 10 positive integers are as follows:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Now we can check whether the given series is A.P. or not by finding common differences.

So common difference ($d$)$ = 2 - 1 = 3 - 2 = 4 - 3 = 1$

As you can see that we have same common difference so the above series is in A.P. and we know the formula for the sum of numbers written in A.P. is as follows:

${{S}_{n}}=\dfrac{n}{2}\left( a+l \right)$ 

In the above formula, “n” is the number of terms, $''a''$ is the first term and $''l''$ is the last term.

Now, the number of terms in the above series is 10 so the value of “n” is 10, the first term in the above series is 1 so the value of $a=1$ and the last term in the above series is 10 so the value of $l=10$. Now, substituting these values in the above summation formula we get,

$ {{S}_{10}}=\dfrac{10}{2}\left( 1+10 \right) $

$ \Rightarrow {{S}_{10}}=5\left( 11 \right) $

$ \Rightarrow {{S}_{10}}=55 $

From the above calculation, we get the sum of the first 10 positive integers is 55.

Note: You might be thinking that we can add the 10 numbers and do the summation why we are using this A.P. formula. You certainly can add all the 10 numbers and proceed but the problem will arise when you have to find the summation for 100 numbers or 1000 numbers then you cannot manually add them so it's always better to use  the A.P. summation formula.