Arithmetic Series

The next topic in our look at arithmetic progressions is arithmetic series. Examine the following A.P.:

2, 5, 8, 11, ...

An arithmetic series is an A.P. where we ADD each term of the A.P. In other words, if you look at the A.P. shown above and replace the commas with plus signs you get:

2+5+8+11+...

Imagine trying to add ALL of the terms in this sequence! You couldn't do it. (It would add up to infinity.) However, you WILL be asked to add up a set number of terms in a series. The formula to help you do this is:

In other words, to find the sum of the first n terms of a series you need to know the first term ( ), the last term () and the number of terms (n). Let's look at an example:

Example

Add the first 29 terms of the A.P. shown here: 3, 11, 19, ...

Solution

One way to solve this problem is just to list the first 29 terms and add them up! This is time consuming, however, and will not help on a test. What if they asked you to add the first 208 terms? We need a better method. Examine the formula that we saw above:

We know that we are looking for the sum of the first 29 terms, therefore n = 29. Putting this into the formula we get:

We already know the value of n, and we know the value of (which is 3). We just need to figure out the value of . To do this, use the formula and method you saw in the lesson on arithmetic term problems. The work is shown below:

So now that we know the value of is 227, we can fill in the rest of the formula used to calculate the sum and solve. We get:

Therefore the sum of the first 29 terms is 3335.

Example

Given the A.P. 8, 2, -4, -10,... find .

Solution

YIKES! What the heck is that thing??? In mathematics, the greek letter sigma () means the sum of or just sum. What the above symbol is telling you to do is find the sum of each term from term 1, (replace the k with 1 in the statement to the right of sigma to get ) to term 18 (again, replacing k with 18 to get ). In other words:

means sum the first 18 terms of the given A.P.

Once you see it written that way, you can probably see the similarity between this example and the last example. We know the value of n (18) and (8). We just need to find the value of . That work is shown here:

Now that we know the value of the first term, the last term, and n, we can use the formula to find the sum of a set number of terms of a sequence:

Just fill in the information you already know and solve.

Therefore, = -774. (Looks ugly doesn't it?)

Now that you have seen some examples involving arithmetic series, you can try some sample questions or head back to the Arithmetic Progressions main page.