Geometric Sequences

We already know that an arithmetic sequence is one where the difference between successive terms is constant. The distance from each term is the same. A geometric sequence is a lot like an arithmetic sequence, but it's completely different at the same time. We can think of it as the doppelgänger of the arithmetic sequence, if we like.

In a geometric sequence, the ratio between successive terms is constant. Geometric sequences grow or shrink at the same ratio from one term to the next. If we divide any two consecutive terms, we'll always get the same ratio.

So if we divide the first terms in a geometric sequence of burritos and get a pinto bean, then dividing burrito-terms two and three of the geometric series will give us a pinto bean of the same size. Let's hope we all like pinto beans.

Sample Problem

Is the sequence 2, 4, 8, 16, … geometric?

Maybe, maybe not. We'll need to take a peek at the ratios of successive terms to find out:

Since these ratios are all the same, the sequence is as geometric as the day is long. And since we've been sitting in class for what feels like an eternity, the day is very long indeed.

Sample Problem

Is the sequence 5, 10, 15, 20, … geometric?

Once again, it's ratio time. Do all the successive terms of our sequence play off the same playbook?

We don't have to go any farther. The ratios are not all the same, so this sequence is not geometric.

Mom always said we should try to keep a positive attitude, but it's okay for the ratio of a geometric series to be negative. The ratio can be anything nonzero.

When the ratio is negative, successive terms change signs, kind of like the flip-flopping snap of a flag waving in the breeze, always pointing one direction and then the other.

Sample Problem

Is the sequence 3, -6, 12, -24, … geometric?

Well, let's take a look at the ratios between the successive terms:

Since the ratios are all -2, this is a geometric sequence.

We mentioned before that although an arithmetic sequence is usually defined in terms of subtraction, we can also think of it in terms of addition. To get from one term to the next, you always add the same thing.

Similarly, while a geometric sequence is defined in terms of division, we can also think of it in terms of multiplication. We said that in a geometric sequence, to get from one term to the previous term, you always divide by the same thing. This is the same as saying that to get from one term to the next, you always multiply by the same thing. This "thing" isn't a swamp monster that has come from the deep to scare little children on Halloween. It is called the common ratio of the sequence and is usually denoted r.

As with the arithmetic sequences, we usually start the geometric sequences off at a₁. If we know a₁ and r, we know everything there is to know about a geometric sequence. We're the masters of the geometric sequence and maybe the swamp monster too.

Sample Problem

A geometric sequence has a₁ = 4 and r = 2. What are the first four terms of the sequence?

We're given a₁, so we're off to a good start. To get the next term (and the next, and the next), we multiply the term we're on by r = 2:

a₂ = a₁(2) = 8

a₃ = a₂(2) = 16

a₄ = a₃(2) = 32

Like eating a piece of cake, we figured out the general formula on our own. To get from a₁ to a_n, we have to multiply by r a total of (n – 1) times. (If it's not quite clear why it's (n – 1) times, go read about steps in arithmetic sequences).

Multiplying by r a total of (n – 1) times is the same thing as multiplying by r^{n – 1}.

If we start at a₁ and multiply by r^{n – 1}, we end up at

a_n = a₁r^{n – 1}.

Because mathematicians need to save their writing hand for scribing infinite sequence strings, it's common to abbreviate a₁ by a. so you'll probably see this formula written

a_n = ar^{n – 1}.

In everything we do with geometric sequences from here on, the letter a refers to the first term of the sequence.

Now that we have this nice formula, if we know a and r for a geometric sequence, we can easily find any term we like.

Just like an arithmetic sequence, if we're given two consecutive terms of a geometric sequence, we can work backward to find r and then a. We know that r is the ratio of successive terms:

Once we know r, we can use the formula a_n = ar^{n – 1} to find a.

Just when we thought we had it all figured out, we have a new problem. How do we find r and a if we have two non-consecutive terms a_m and a_n? This is no simple arithmetic sequence. Some people might throw their hands up in the air and go eat lunch. We're going to flex our brain muscles instead of our stomachs.

This is no impossible labyrinth to find our way out of. We only need to perform one extra operation to solve this problem, but it requires knowing how many times we multiply by r to get from a_m to a_n (assume m < n).

Let's look at our equation for a geometric sequence. The n^th term is a_n = ar^{n – 1}, while the m^th term is a_m = ar^{m – 1}.

Let's divide these equations:

We can cancel the a out, and we will multiply by , which is just 1 in disguise. Sneaky devil. This gives us:

Rearranging this equation so that we have r all on its lonesome, we get:

So, once we know how many steps we need to take, we have (n – m). We divide a_n by a_m, and then we take the (n – m)th root. That gives us r. Then we proceed as before to find a. Does your brain feel bigger yet?

1) Find .

2) Take the (n – m)^th root to get r.

3) Use the geometric sequence formula to find a.

Remember that we should always check our answers by using the values of a and r we found in the formula. If we found the right values, we should be able to use the formula to get the terms a_m and a_n that we started with.

As with an arithmetic sequence, a geometric sequence is completely determined by two pieces of information. If you know a and r, you know everything there is to know about the sequence. If you know any two terms a_m and a_n, you can find r and a, which means you're the master of the geometric sequence.