Geometric Sequences and the Geometric Sequence Formula

A geometric sequence is an ordered list of numbers in which each term after the first is found by multiplying the previous term by a fixed number called the common ratio, usually written as \(r\). For example, in the sequence \(3, 6, 12, 24, 48, \ldots\), each term is double the one before it, so the common ratio is \(r = 2\).

To check whether a sequence is geometric, divide any term by the term that comes right before it. If that ratio stays the same all the way through the sequence, it is geometric.

\(r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \ldots\)

The most useful tool for working with geometric sequences is the formula for the nth term, sometimes called the explicit formula:

\(a_n = a_1 \cdot r^{\,n-1}\)

Here, \(a_n\) is the term you want to find, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence. This formula lets you jump straight to any term, such as the 20th or 100th, without listing every term in between.

Worked Example 1: Finding a Specific Term

Find the 7th term of the geometric sequence \(5, 15, 45, 135, \ldots\)

First identify \(a_1 = 5\). The common ratio is \(r = \frac{15}{5} = 3\). Substitute into the formula:

\(a_7 = 5 \cdot 3^{\,7-1} = 5 \cdot 3^6 = 5 \cdot 729 = 3645\)

Worked Example 2: Writing the Explicit Formula

Write the explicit formula for the sequence \(80, 40, 20, 10, \ldots\)

Here \(a_1 = 80\) and \(r = \frac{40}{80} = \frac{1}{2}\). So the formula is:

\(a_n = 80 \cdot \left(\frac{1}{2}\right)^{n-1}\)

Notice that when \(0 < |r| < 1\), the terms shrink toward zero rather than growing, as shown below.

Instead of jumping directly to a term, the recursive formula defines each term using the term immediately before it:

\(a_n = a_{n-1} \cdot r, \qquad n \ge 2\)

This says: to get the next term, multiply the current term by the common ratio. The recursive formula always needs a starting value, \(a_1\), before it can be used, since every later term is built from it step by step.

Worked Example 3: Using the Recursive Formula

Given \(a_1 = 4\) and \(a_n = a_{n-1} \cdot 5\), find the first four terms.

\(a_1 = 4\), \(a_2 = 4 \cdot 5 = 20\), \(a_3 = 20 \cdot 5 = 100\), \(a_4 = 100 \cdot 5 = 500\). The sequence is \(4, 20, 100, 500, \ldots\)

It helps to compare geometric sequences with arithmetic sequences, since students often mix the two up. In an arithmetic sequence, you add the same fixed number to get the next term, so the terms grow at a constant rate. In a geometric sequence, you multiply by the same fixed ratio, so the terms grow or shrink exponentially. A quick way to tell them apart: check whether consecutive terms have a common difference (arithmetic) or a common ratio (geometric).

A geometric sequence lists the individual terms, but once you start adding those terms together you get a geometric series. When the common ratio has an absolute value less than 1 and the sum continues forever, that sum can still settle on a finite value, which is the idea behind an infinite geometric series. Those topics build directly on the common ratio and nth term ideas covered here.

Watch out for these frequent errors when working with geometric sequences:

• Subtracting terms instead of dividing when looking for the common ratio (that method finds a common difference, which only applies to arithmetic sequences).
• Forgetting the exponent is \(n - 1\), not \(n\), in the explicit formula.
• Mishandling a negative common ratio, which makes the terms alternate in sign, for example \(2, -6, 18, -54, \ldots\)
• Assuming a sequence is geometric just because the numbers are growing, without actually checking that the ratio between consecutive terms is constant.

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