Use a Formula for a Geometric Sequence

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 $r$ and first term $a_1$ is

$a_n=ra_{n−1}, n \geq 2$

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 following geometric sequence.

$\{6, 9, 13.5, 20.25, ...\}$

Solution

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

$r=\frac{9}{6}=1.5$

Substitute the common ratio into the recursive formula for geometric sequences and define $a_1$.

$\begin{aligned}&a_{n}=r a_{n-1} \\&a_{n}=1.5 a_{n-1} \text { for } n \geq 2 \\&a_{1}=6\end{aligned}$

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

Q&A

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

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.

Try It #4

Write a recursive formula for the following geometric sequence.

$\left\{2, \frac{4}{3}, \frac{8}{9}, \frac{16}{27}, \cdots\right\}$

Source: Rice University, https://openstax.org/books/college-algebra/pages/9-3-geometric-sequences
This work is licensed under a Creative Commons Attribution 4.0 License.

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.

$a_n=a_1r^{n−1}$

Let's take a look at the sequence $\{18, 36, 72, 144, 288, ...\}$. 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

$a_n=18⋅2^{n−1}$

The graph of the sequence is shown in Figure 3.

Figure 3

Explicit Formula for a Geometric Sequence

The $nth$ term of a geometric sequence is given by the explicit formula:

$a_n=a_1r^{n−1}$

Example 4

Writing Terms of Geometric Sequences Using the Explicit Formula

Given a geometric sequence with $a_1=3$ and $a_4=24$, find $a_2$.

Solution

The sequence can be written in terms of the initial term and the common ratio $r$.

$3,3r,3r^2,3r^3,...$

Find the common ratio using the given fourth term.

$\begin{aligned} &a_{n}=a_{1} r^{n-1}\\ &a_{4}=3 r^{3} & & \text { Write the fourth term of sequence in terms of } \alpha_{1} \text { and } r\\ &24=3 r^{3} & & \text { Substitute } 24 \text { for } a_{4}\\ &8=r^{3} & & \text { Divide }\\ &r=2 & & \text { Solve for the common ratio } \end{aligned}$

$\begin{aligned} a_2 &= 2a_1 \\ &=2(3) \\ &=6 \end{aligned}$

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.

Try It #5

Given a geometric sequence with $a_2=4$ and $a_3=32$, find $a_6$.

Example 5

Writing an Explicit Formula for the $n th$ Term of a Geometric Sequence

Write an explicit formula for the $nth$ term of the following geometric sequence.

Solution

The first term is 2. The common ratio can be found by dividing the second term by the first term.

$\frac{10}{2}=5$

The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.

$\begin{aligned} a_n &=a_1r^{n-1} \\ a_n &=2\cdot5^{n-1} \end{aligned}$

Figure 4

Try It #6

Write an explicit formula for the following geometric sequence.

$\{–1, 3, –9, 27, ...\}$

In real-world scenarios involving geometric sequences, we may need to use an initial term of $a_0$ instead of $a_1$. In these problems, we can alter the explicit formula slightly by using the following formula:

$a_n=a_0r^n$

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.

Solution

Try It #7

A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.