Use a Formula for a Geometric Sequence

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.

$\{–1, 3, –9, 27, ...\}$