Write the Terms of a Geometric Sequence

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.

1. Multiply the initial term, $a_1$, by the common ratio to find the next term, $a_2$.
   
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.
   
3. 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.

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

Try It #3

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