# Geometric Sequences

A series of free, online lessons for Intermediate Algebra (Algebra II) with videos, examples and solutions.

In these lessons, we will learn

• how to find the common ratio of a geometric sequence
• how to find the formula for the nth term of an geometric sequence
• how to find the sum of an geometric series

The following figure gives the formula for the nth term of a geometric sequence. Scroll down the page for more examples and solutions. Geometric Sequences

A geometric sequence is a sequence that has a pattern of multiplying by a constant to determine consecutive terms.

We say geometric sequences have a common ratio.
The formula is
an = an-1r

Examples:

1. A sequence is a function. What is the domain and range of the following sequence? What is r?
-12, 6, -3, 3/2, -3/4

2. Given the formula for the geometric sequence, determine the first 2 terms and then the 5th term. Also state the common ratio.

3. Given the geometric sequence, determine the formula. Then determine the 6th term.
1/3, 2/9, 4/27, 8/81, …

A Quick Introduction To Geometric Sequences

This video gives the definition of a geometric sequence and go through 4 examples, determining if each qualifies as a geometric sequence or not.

A geometric sequence is a sequence of numbers where each term after the first term is found by multiplying the previous one by a fixed non-zero number, called the common ratio.

Example: Determine which of the following sequences are geometric. If so, give the value of the common ratio, r.

1. 3,6,12,24,48,96, …

2. 3,3/2,3/4,3/8,3/16,3/32,3/62, …

3. 10,15,20,25,30, …

4. -1,.1,-.01,-.001,-.0001, …

Geometric Sequences

A list of numbers that follows a rule is called a sequence. Sequences whose rule is the multiplication of a constant are called geometric sequences, similar to arithmetic sequences that follow a rule of addition. Homework problems on geometric sequences often ask us to find the nth term of a sequence using a formula. Geometric sequences are important to understanding geometric series.

How To Find The General Term Of A Geometric Sequence?

Example:
Find the formula for the general term or nth term of a geometric sequence.

Geometric Sequences And Series

A short introduction to geometric sequences and series.

Math Skills & Equations: Solving Math Sequences

There are two kinds of math sequences that can be solved: arithmetic sequences and geometric sequences. An arithmetic sequence is solved by adding or subtracting the same number, while geometric sequences use division and multiplication. Learn more about solving math sequences.

Example:
Find the 9th term of 3,12,48,192, …

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 