Simple Steps to Find the Sum of an Arithmetic Series

An arithmetic series is when you add up the numbers in an arithmetic sequence. In this kind of sequence, each number gets bigger by the same amount each time.

The important formula to know is:

$S_n = \frac{n}{2} (a_1 + a_n)$

Here's what those letters mean:

• $S_n$ is the sum of the first $n$ numbers.
• $a_1$ is the first number in the sequence.
• $a_n$ is the last number you're adding, or the $n^{th}$ number.
• $n$ is the total number of terms.

Step 1: Find the First Term and Common Difference

First, look at your arithmetic sequence:

• Let’s call the first term $a_1$.
• The difference between each term will be called $d$.

The sequence looks like this:

• $a_1, a_2, a_3, \ldots, a_n$

You can also find any term using:

• $a_k = a_1 + (k-1)d$ for $k = 1, 2, \ldots, n$.

Step 2: Write Down the Last Term

To find the last term ($a_n$) in the series:

$a_n = a_1 + (n-1)d$

Step 3: Add Up the Series

Now, let’s write out the sum of the first $n$ numbers:

$S_n = a_1 + a_2 + a_3 + \ldots + a_n$

Step 4: Write the Series Backwards

Next, let’s write the same series but backwards:

$S_n = a_n + a_{n-1} + a_{n-2} + \ldots + a_1$

Step 5: Combine the Two Lists

Now, we will add these two lists together:

$$2S_n = (a_1 + a_n) + (a_2 + a_{n-1}) + (a_3 + a_{n-2}) + \ldots + (a_n + a_1)$$

Notice that when you add the numbers in pairs, they all equal the same amount:

$2S_n = n(a_1 + a_n)$

Step 6: Find the Formula for $S_n$

To get the final answer for $S_n$, divide both sides by 2:

$S_n = \frac{n}{2} (a_1 + a_n)$

Final Formula

So, the formula for the sum of the first $n$ terms in an arithmetic series is:

$S_n = \frac{n}{2} (a_1 + a_n)$

Key Points to Remember

1. The formula shows how the sum of an arithmetic series is based on the number of terms, the first term, and the last term.
2. You can use this formula for any specific values of $n$, $a_1$, or $d$.
3. Understanding this formula is useful for solving problems related to arithmetic sequences, especially in Grade 12 Pre-Calculus.