How to Compute the Sum of first n terms?

Given n and an AP, how do we calculate the sum of the first n terms? Let’s derive the formula for it:
Let S_n denote the sum for the first n terms, we can write S_n in two ways:

You see both are the same but written in opposite directions, i.e. in the first elements are from the first term to the last term, and in the second the elements are from last to first. 
Now add the above two:-
 

Computing Sum of first n natural numbers

Problem: Calculate the sum of natural numbers from 1 to 100. 
 

According to anecdotes, when Gauss was in primary school, he was punished by his teacher due to misbehavior.  Gauss was told to add the numbers from 1 to 100 (the teacher gave that considering that a tedious job). But he was able to compute its sum, which is 5050, in a matter of seconds.  How did he do?
He did something like: 1 + 2 + 3 + . . . . + 49 + 50 + (100 – 49) + . . . . + (100 – 2) + (100 – 1) + 100
Which cancels out to 49 * 100 + 100 + 50 = 50 * 100 + 50 = 50(100 + 1). Gauss was great, but we aren’t less though, since we have our formula, we will use that.

Solution: We see the natural numbers (1, 2, 3, …) form an AP and hence we can calculate the sum of the first n terms as follows:
First-term, a = 1
Difference, d = 1
So, 

Sn = (n/2) × [2a + (n - 1) × d]
Sn = (n/2) × [2 × 1 + (n - 1) × 1]
Sn = (n/2) × (n + 1) 

For n = 100, we have:

S100 = (100/2) × (100 + 1)
S100 = 5050

Now you are already good at Arithmetic Progressions!

In Algebra a term is either a single number or variable or numbers and variables multiplied together. Terms are separated by + or − signs, or sometimes by the divide. 
Examples: In expression 5x – 10 = 1, 5x, 10 and 1 are the terms.