Finite Arithmetic Series Formula

Happy Wednesday math peeps! Today we are going to go over the Finite Arithmetic Series Formula; What it is, how to use it, and even do a little derivation. Before going any further though, please make sure you know how arithmetic sequences work here. If you have any questions, please don’t hesitate to comment below and to 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.

Example:

But when what happens if we wanted to sum the terms of our arithmetic sequence together?

Example:

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?  That’s where our Finite Arithmetic Series formula comes in handy!

Finite Arithmetic Series Formula:

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:

*Bonus* Arithmetic Series Formula:

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/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

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, ….

Solutions:

1) 480

2) 1,428

3) 1,712

4) 2,700

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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Looking to learn about Finite Geometric Series? Check out this post here!

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