Sequences - I - The Arithmetic Sequence

Many college math texts assume that you clearly remember the nature of sequences and how the formulas for handling them were developed when you took a basic algebra course. My experience tells me that most of you only have a vague recollection of sequences and have no idea how the formulas are derrived. So, this section will start with the basics on arithmetic sequences and go through the process of working with them. The following section will deal with geometric sequences. If you have both of these down pat you will not get much out of this section of the website.

If you are still reading you probably don't remember much about arithmetic sequences or may not have even come across them before.

An arithmetic sequence is a sequence where you have a common (constant) difference between successive terms.

Individual Terms in a Sequence

When sequences are talked about in a general way, with symbols the terms of the sequence are describes as a1, a2... The difference between terms, the common difference is usually represented by the letter d. So you would say that d = an+1 - an. In other words d is equal to the difference between two adjacent terms in the sequence. If we are working with a sequence we can write the terms as follows:

Let's use what was discussed above. We are given the sequence 7, 3, -1, -5, ... The question then asks us to find the common difference.

To help solidify this let's look at a few problems. We'll start simply and progress to the (impossibly) difficult.

Determining the general expression when given a sequence



Sum of the Terms in a Sequence

Up to this point we have discussed the individual terms of a sequence. The next question to address is the summing of all those individual terms to get the sum of the terms in a sequence. Let me first introduce the Greek capital letter sigma written as Σ. When you see this letter it should tell you to sum all of the terms that follow it. Usually it is written with a lower limit, on the bottom and an upper limit, on the top. Here is an example (yes, its tiny and hard to read). This tells us that i goes from 1 to 5 inclusive and asks us to add the terms 1, 2, 3, 4 and 5. In other words it says total 1 + 2 + 3 + 4 + 5. The answer being 15. We have found the sum of an arithmetic sequence of 5 terms. The first term a1 was equal to 1 and the difference, d, was equal to 1.

Lets next consider an arithmetic sequence that goes from 1 to 10 with a difference between terms of 1.

To start let's look at a problem that asks us to sum the first thirty terms of the arithmetic sequence 3, 7, 11, 15 ..

  • ..

    Let's look at a slightly different version of the problem. Suppose we are given the sequence 7 + 10 + 13 + ... + 157 and asked to find the sum of the sequence. We know the first term is 7 the last term is 157 and the difference between terms in 3 but we don't know n the number of terms in the sequence. How can we find n? Well we have a formula that says an = a1 + (n-1)*d and we know everything in this formula except n. So we can use it to find n.



    • Determining the sum of the terms of a sequence



    Word Problems

    Let's end this page with a couple of word problems.