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 \(r=\frac{1}{3}\).