Geometric Sequences and Series
A culture of bacteria that doubles every hour, a loan balance that grows by a fixed percent each month, and a bouncing ball that returns to a fixed fraction of its last height all follow the same rule: multiply by the same number to get the next term. That fixed multiplier is what makes a sequence geometric, and it lets you jump straight to the 100th term or add up a long run of terms without listing them.
By the end of this page you can:
- Recognize a geometric sequence and find its common ratio.
- Write the nth term from the first term and the ratio.
- Sum a finite geometric series with the closed formula.
- Decide when an infinite geometric series converges and find its sum.
The floor. Each term is the previous term times a fixed number r. The nth term is a_1 times r raised to the (n minus 1) power, and a finite sum has a one-line formula.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7e |
| Chapter | 12 |
| Section | 12.3 Geometric Sequences |
| Open alternate | OpenStax Precalculus 2e, 11.3 Geometric Sequences |
| Difficulty | Core |
| Time | 30 minutes |
Before You Start
The Idea
A sequence is geometric when the ratio of each term to the one before it is the same constant. That constant is the common ratio r. To test a sequence, divide any term by the previous term: if you always get the same number, the sequence is geometric and that number is r. To build the sequence, you start at a_1 and multiply by r over and over.
Because every step multiplies by r, reaching the nth term means multiplying a_1 by r a total of (n minus 1) times. The growth is multiplicative, not additive, which is the key contrast with an arithmetic sequence: an arithmetic sequence adds a common difference, a geometric sequence multiplies by a common ratio.
Key Formulas
The nth term of a geometric sequence with first term a_1 and common ratio r:
\[ a_n = a_1\, r^{\,n-1} \]
The sum of the first n terms (a finite geometric series), valid when r is not 1:
\[ S_n = a_1\,\frac{1 - r^{\,n}}{1 - r} \]
The sum of an infinite geometric series, valid only when the ratio satisfies the size condition:
\[ S = \frac{a_1}{1 - r}, \qquad \text{provided } |r| < 1 \]
If |r| is greater than or equal to 1, the infinite series does not converge and has no finite sum.
Source: Stewart 7e, Section 12.3 Geometric Sequences.
Worked Example
Find the 6th term and the sum of the first 6 terms of the geometric sequence 3, 6, 12, 24, ...
Step 1. Find the common ratio. Divide a term by the one before it: 6 divided by 3 is 2, and 12 divided by 6 is 2. So r is 2 and a_1 is 3.
Step 2. Find the 6th term using a_n = a_1 r^(n-1) with n = 6.
Step 3. Find the sum of the first 6 terms using the finite-series formula with the same a_1, r, and n.
Show the answer
Sixth term:
\[ a_6 = 3 \cdot 2^{\,6-1} = 3 \cdot 2^5 = 3 \cdot 32 = 96 \]
Sum of the first six terms:
\[ S_6 = 3 \cdot \frac{1 - 2^{6}}{1 - 2} = 3 \cdot \frac{1 - 64}{-1} = 3 \cdot \frac{-63}{-1} = 3 \cdot 63 = 189 \]
Check by adding directly: 3 + 6 + 12 + 24 + 48 + 96 = 189. The formula matches the hand count.
Connections
This builds on sequences and summation notation, where you learned to read a general term and write a sum compactly. It pairs with arithmetic sequences: arithmetic adds a common difference, geometric multiplies by a common ratio, and keeping the two formulas separate is the main thing to guard against. A common error is reaching for the arithmetic sum formula on a geometric series, or forgetting that the infinite sum exists only when the absolute value of the ratio is less than 1.
Looking forward, the closed sum formula here is a natural target for a proof by mathematical induction, where you verify the formula for the first case and then show each step forces the next. Geometric series also reappear in calculus as the model for convergence.
Resources
- Stewart, Redlin, Watson, Precalculus 7e, Section 12.3 Geometric Sequences (primary).
- OpenStax Precalculus 2e, Section 11.3 Geometric Sequences (open alternate).
Last updated: 2026-06-24