The formula to find the sum of an arithmetic series is:

$$S_n = \frac{n}{2} (a + l)$$

or

$$S_n = \frac{n}{2} (2a + (n - 1)d)$$

Here's what the letters mean:

• $S_n$: This is the sum of the first $n$ terms.
• $a$: The first term of the series.
• $l$: The last term of the series.
• $d$: The common difference between the terms.
• $n$: The total number of terms.

How to Understand the Formula

1. What is an Arithmetic Series?
   An arithmetic series is created by adding the numbers in an arithmetic sequence. In this sequence, each number increases by a fixed amount called the common difference ($d$). For example, in the sequence $2, 5, 8, 11$, the first term is $2$ and the common difference is $3$ (because $5 - 2 = 3$).

2. Writing the Sum of the Terms
   To find the sum of the first $n$ terms, we can write it like this:
   $S_n = a + (a + d) + (a + 2d) + \ldots + (a + (n-1)d)$

3. Writing it Backward
   Now, let’s write the same sum in reverse order:
   $S_n = (a + (n-1)d) + (a + (n-2)d) + \ldots + a$

4. Combining Both Sums
   If we add these two sums together, we get:
   $2S_n = (a + l) + (a + l) + \ldots + (a + l)$
   This means each pair of terms adds up to $a + l$, and there are $n$ pairs like that.

5. Final Simplification
   So, we can simplify it to:
   $2S_n = n(a + l)$
   This leads us to the formula:
   $S_n = \frac{n}{2}(a + l)$

This formula is a simple way to quickly find the sum of an arithmetic series. It shows up a lot in math and helps in many calculations.