Write the Terms of a Geometric Sequence

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 $a_1=−2$ and the common ratio is $r=4$ , we can find subsequent terms by multiplying $−2⋅4$ to get $−8$ then multiplying the result $−8⋅4$ to get $−32$ and so on.

$\begin{array}{ll}a_1 = -2 \\a_2 = (-2\cdot4) = -8 \\a_3 = (-8\cdot4) = -32 \\a_4 = (-32\cdot4) = -128 \\\end{array}$

The first four terms are $\{–2, –8, –32, –128\}$ .

How To

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

Multiply the initial term, $a_1$ , by the common ratio to find the next term, $a_2$ .
Repeat the process, using $a_n=a_2$ to find $a_3$ and then $a_3$ to find $a_4$ , until all four terms have been identified.
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 $a_1=5$ and $r=–2$ .

Solution

Multiply $a_1$ by $−2$ to find $a_2$ . Repeat the process, using $a_2$ to find $a_3$ , and so on.

$\begin{array}{ll}a_1 = 5 \\a_2 = -2a_1 = -10 \\a_3 = -2a_2 = 20 \\a_4 = -2a_3 = -40 \\\end{array}$

The first four terms are

$\{5,–10,20,–40\}$ .

Try It #3

List the first five terms of the geometric sequence with $a_1=18$ and $r=\frac{1}{3}$ .