Finite Arithmetic Sequence

Happy Wednesday math peeps! Today we are going to go over the Finite Arithmetic Sequence; What is the finite arithmetic sequence? How do we use it? And where did it come from? Before going any further though, please make sure you know how arithmetic sequences work before reading onward. If you have any questions, please don’t hesitate to comment below and check out the video and practice questions below. Happy calculating! 🙂

What does it mean to find the “Sum of the Finite Arithmetic Sequence”?

We already know what an arithmetic sequence is: a sequence of numbers that forms a pattern when the same number is added or subtracted to each term. Check out the example below where we are adding the same number, 2 (known as the common difference), to each subsequent term of the sequence.


common ratio

But what happens now if we wanted to sum the terms of our arithmetic sequence together? Adding 2+4+6+8+……


Arithmetic Series

More specifically, what if we wanted to find the sum of the first 20 terms of the above arithmetic sequence?  How would we calculate that? Instead of finding the first 20 terms of our sequence and adding them all together, thankfully there is a better way, which is where our Finite Arithmetic Series formula comes in handy!

Finite Arithmetic Series Formula

Why is it called “finite”? Adding the first 20 terms of our arithmetic sequence are considered to be “finite” because the first 20 terms have a definite ending as opposed to a sequence that is infinite and goes on forever. Adding together an infinite series comes with a different formula.

Finite Arithmetic Series Formula:

a1=The first term of our sequence. In this case, we can see that the first term will be the number 2 in the example above. Therefore, we can say a1=2.

an= Value of the last term, in the case mentioned above where we want to find the sum of the first 20 terms, this would be the value of the 20th term.

n= The total number of terms we are trying to sum together. In the example mentioned above, we are trying to sum 20 terms in total, so in this case n=20.

Finite Arithmetic Series Formula

Looking at the above formula, I have to wonder, what happens if we are not given the value of the last term of the sequence for “a sub n”?  What would we do? Do not worry, because there is another way to use this formula if we expand and simplify it, check it out below:

Arithmetic Series Formula: Where did it Come From?

Plug in the arithmetic sequence formula for “a sub n,” then combine like terms.

Let’s take a closer look at what each part of our bonus formula represents below:

Finite Arithmetic Series Formula

Now that we have two formulas to work with, let’s take another look at our question now applying our finite arithmetic series formula:

Step 1: First let’s write out our formula and identify what each part represents and what numbers need to be filled in. Since we are not given the value of the last term, “a sub n” we can use the second bonus formula we previously derived.

Finite Arithmetic Series Formula

Step 2: Now let’s fill in our formula and calculate.

Finite Arithmetic Series Formula

Think you are ready to try some finite arithmetic sequence questions on your own? Test your skills with the following finite arithmetic progression practice examples below:

Practice Questions:

1) Find the sum of the first 15 terms of the following arithmetic sequence:

4, 8, 12, 16, ….

2) Find the sum of the first 24 terms of the following sequence:

2, 7, 12, 17, ….

3) Find the sum of the first 32 terms of the following arithmetic sequence:

100, 97, 94, 91, ….

4) Find the sum of the first 50 terms of the following arithmetic sequence:

5, 7, 9, 11, 13, ….


1) 480

2) 1,428

3) 1,712

4) 2,700

Related Posts:

Looking to learn more about sequences? You’ve come to the right place! Check out these sequence resources and posts below. Personally, I recommend looking at the finite geometric sequence or the infinite geometric series posts next!

Geometric Sequence

Recursive Formula

Arithmetic Sequence

Finite Geometric Series

Infinite Geometric Series

Golden Ratio in the Real World

Fibonacci Sequence

Still, got questions? No problem! Don’t hesitate to comment with any questions below or check out the video above. Thanks so much for stopping by and happy calculating! 🙂

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