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Geometric Sequences

Jay Abramson

Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26 000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26 520 after one year; $27 050.40 after two years; $27 591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.

1. Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term. We visualize this as follows:

Definition of a Geometric Sequence

A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If is the initial term of a geometric sequence and is the common ratio, the sequence will be

How To …

Given a set of numbers, determine if they represent a geometric sequence.

  1. Divide each term by the previous term.
  2. Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.

Example 1: Finding Common Ratios

Is the sequence geometric? If so, find the common ratio.

Solution

Divide each term by the previous term to determine whether a common ratio exists.

  1. We have

    The sequence is geometric because there is a common ratio. The common ratio is 2.

  2. We have

    The sequence is not geometric because there is not a common ratio.

Analysis

The graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.

Figure 1: (a) A graph representing the first five terms of a geometric sequence. (b) A graph representing the first five terms of a sequence which is not geometric.

Question

If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?

Answer

No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.

2. Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is and the common ratio is , we can find subsequent terms by calculating to get then calculating to get and so on. Namely, and so on. Thus, the first four terms are .

How To …

Given the first term and the common factor, find the first four terms of a geometric sequence.

  1. Multiply the initial term, , by the common ratio to find the next term, .
  2. Repeat the process, using to find and then to find , until all four terms have been identified.
  3. Write the terms separated by commons within brackets.

Example 2: Writing the Terms of a Geometric Sequence

List the first four terms of the geometric sequence with and .

Solution

Multiply by to find . Repeat the process, using to find , and so on. Namely,

Thus, the first four terms are .

3. Using Recursive Formulas for Geometric Sequences

A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

Recursive Formula for a Geometric Sequence

The recursive formula for a geometric sequence with common ratio and first term is for .

How To …

Given the first several terms of a geometric sequence, write its recursive formula.

  1. State the initial term.
  2. Find the common ratio by dividing any term by the preceding term.
  3. Substitute the common ratio into the recursive formula for a geometric sequence.

Example 3: Using Recursive Formulas for Geometric Sequences

Write a recursive formula for the geometric sequence

Solution

The first term is given as 6. The common ratio can be found by dividing the second term by the first term. Hence we find

Substituting the common ratio into the recursive formula for geometric sequences we get for . It only remains to specify that .

Analysis

The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.

Figure 2: A graph showing the first four terms of the geometric sequence with recursive formula for , where .

Question

Do we have to divide the second term by the first term to find the common ratio?

Answer

No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.

4. Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms. For example

Let’s take a look at the sequence . This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is for .

The graph of the sequence is shown in Figure 3.

Figure 3: A graph showing the first five terms of the sequence with formula .

Explicit Formula for a Geometric Sequence

The th term of a geometric sequence is given by the explicit formula: for .

Example 4: Writing Terms of Geometric Sequences Using the Explicit Formula

Given a geometric sequence with and , find .

Solution

The sequence can be written in terms of the initial term and the common ratio . Namely,

Substituting for the first and fourth term in the formula we get

Solving for we find .

We find the second term by multiplying the first term by the common ratio.

Thus,

Analysis

The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.

Example 5: Writing an Explicit Formula for the th Term of a Geometric Sequence

Write an explicit formula for the th term of the geometric sequence

Solution

The first term in the sequence is 2. The common ratio can be found by dividing the second term by the first term. So calculating we find that the commom ratio is 5. Substituting the common ratio and the first term of the sequence into the formula, we get

The graph of this sequence in Figure 4 shows an exponential pattern.

Figure 4: A graph showing the first four terms of the sequence with formula .

5. Solving Application Problems with Geometric Sequences

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of instead of . In these problems, we can alter the explicit formula slightly by using the following formula: where .

Example 6: Solving Application Problems with Geometric Sequences

In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.

  1. Write a formula for the student population.
  2. Estimate the student population in 2020.

Solution

  1. The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.

    Let be the student population and be the number of years after 2013. Using the explicit formula for a geometric sequence we get

  2. We can find the number of years since 2013 by subtracting as follows:

    We are looking for the population after 7 years. We can substitute 7 for to estimate the population in 2020. We find that Thus, the student population will be about 374 in 2020.