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High School Precalculus: Help and Review32 chapters | 282 lessons

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Instructor:
*David Liano*

After completing this lesson, you will be able to recognize a geometric sequence. You will also be able to use the general formula for finding a term in a geometric sequence and will be able to write a custom formula for a given geometric sequence.

In mathematics, a **sequence** is usually meant to be a progression of numbers with a clear starting point. What makes a sequence geometric is a common relationship that exists between any two consecutive numbers in the sequence.

Let's consider the NCAA basketball tournament. After the preliminary rounds, the tournament has a field of 64 teams. In the round of 64, all teams play, so there will be 32 teams eliminated. In other words, there are 32 teams left, or half of what we started with. After the round of 32, there are 16 teams left. Again the number of teams has been cut in half.

This pattern continues until there is one team left. Let's write this as a sequence:

64, 32, 16, 8, 4, 2, 1

Do you see the relationship between any two consecutive terms? Each term after the first term is ½ of the preceding term. Another way to look at it is that we are multiplying each term by ½ to get the next term in the sequence. Also notice that the ratio of any term and its preceding term is ½. For example 32/64 = ½ and 2/4 = ½. This is called the **common ratio** of the geometric series, and it is denoted by *r*. This ratio must hold true for any pair of consecutive terms. Otherwise, the sequence is not a geometric sequence.

This example is a **finite geometric sequence**; the sequence stops at 1. Some geometric sequences continue with no end, and that type of sequence is called an **infinite geometric sequence**.

Let's look at other examples of geometric sequences:

6, 12, 24, 48, 96, ...

4, -6, 9, -13.5, ...

The first sequence has a common ratio of 2, as is shown next:

12/6 = 24/12 = 48/24 = 96/48 = 2

The second sequence is also geometric. It might be hard to see at first, but it does have a common ratio of (-3/2) as is shown next:

-6/4 = 9/-6 = -13.5/9 = -3/2

Let's now look at some sequences that are not geometric:

1, 4, 9, 16, 25, ...

100, 90, 80, 70, 60, ...

In each sequence, the ratio between consecutive terms is not the same. For instance, 4/1 does not equal 9/4 in the first sequence. In the second sequence, 90/100 does not equal 80/90.

The *n*th term of a geometric sequence is identified as *a*(*n*). For instance, *a*(1) is the first term of the sequence, and *a*(7) is the seventh term of the sequence. To get from one term of a sequence to the next, we need to multiple the preceding term by the common ratio *r*. The rule for finding the *n*th term of a sequence is shown in Figure 1.

Notice that the first term *a*(1) is multiplied by *r* to the power of (1 - 1) or zero. Any number to the power of zero is 1, so we are just multiplying the first term by 1. As we calculate each next term, we just keep multiplying by *r*. The seventh term would be *a*(1) multiplied by *r* six times or *r*^6.

Let's write a rule for the *n*th term of the following geometric sequence:

3, 15, 75, 375, 1,875, …

The first term is *a*(1) = 3. The common ratio *r* formed by using any pair of consecutive terms is 15/3 = 5. We can substitute these values into the general rule for a geometric sequence as shown in Figure 2:

Now that we have a rule for this sequence, we can easily find any term of the sequence. Let's find *a*(9):

*a*(9) = 3(5)^(9 - 1)

*a*(9) = 3(5)^8

*a*(9) = 1,171,875

Let's write a rule for the *n*th term of a geometric sequence with a common ratio of 6 and *a*(3) = 72.

We are given *r*, but we need to find *a*(1).

*a*(*n*) = *a*(1)r^(*n* - 1) (write general rule)

*a*(3) = *a*(1)*r*^(3-1) (replace *n* with 3)

72 = *a*(1)6^2 (replace *a*(3) and *r*)

2 = *a*(1) (solve for *a*(1))

We can now write the rule shown in Figure 3.

A geometric sequence is formed by multiplying each preceding term by the same factor, which we call the common ratio *r*. When given the rule for a specific geometric sequence, the first term *a*(1) of the sequence and the common ratio *r* of the sequence can be clearly identified. When not given the rule for a specific geometric sequence, the rule can be found with just a few calculations, given the appropriate information.

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