Use a Formula for a Geometric Sequence

Finally, we will find the terms of a geometric sequence given a recursive formula.

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 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
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