88 Geometric Sequences
Learning Objectives
In this section, you will:
- Find the common ratio for a geometric sequence.
- List the terms of a geometric sequence.
- Use a recursive formula for a geometric sequence.
- Use an explicit formula for a geometric sequence.
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.
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.
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 [latex]{a}_{1}[/latex] is the initial term of a geometric sequence and [latex]r[/latex] is the common ratio, the sequence will be
How To
Given a set of numbers, determine if they represent a geometric sequence.
- Divide each term by the previous term.
- Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.
Finding Common Ratios
Is the sequence geometric? If so, find the common ratio.
- [latex]1\text{,}\,2\text{,}\,4\text{,}\,8\text{,}\,16\text{,}\,...[/latex]
- [latex]48\text{,}\,12\text{,}\,4\text{, }2\text{,}\,...[/latex]
[hidden-answer a=”fs-id1165137430766″]Divide each term by the previous term to determine whether a common ratio exists.
- [latex]\begin{array}{llllllllll}\frac{2}{1}=2\hfill & \hfill & \hfill & \frac{4}{2}=2\hfill & \hfill & \hfill & \frac{8}{4}=2\hfill & \hfill & \hfill & \frac{16}{8}=2\hfill \end{array}[/latex]
The sequence is geometric because there is a common ratio. The common ratio is 2.
- [latex]\begin{array}{lllllll}\frac{12}{48}=\frac{1}{4}\hfill & \hfill & \hfill & \frac{4}{12}=\frac{1}{3}\hfill & \hfill & \hfill & \frac{2}{4}=\frac{1}{2}\hfill \end{array}[/latex]
The sequence is not geometric because there is not a common ratio.
[/hidden-answer]
Analysis
The graph of each sequence is shown in (Figure). It seems from the graphs that both (a) and (b) appear have the form of the