Geometric Sequences

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

In this lesson, we will learn

• about geometric sequences
• 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.

 

Sequences Worksheets
Practice your skills with the following worksheets:
Printable & Online Sequences Worksheets

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 for the n-th term (a_n) of a geometric sequence is:
a_n = a₁r^n-1
Where:
a_n is the n-th term
a₁ is the first term
r is the common ratio
n is the term number (position in the sequence)

Sum of a Finite Geometric Sequence
The sum of the first n terms of a geometric sequence (S_n) is given by:
\(S_n=\frac{a_{1}\left( 1-r^n \right)}{1-r}\) for r ≠ 1
if r = 1, the sum is S_n = na₁

Sum of an Infinite Geometric Sequence
For an infinite geometric sequence, the sum (S_∞) only converges (approaches a finite value) if the absolute value of the common ratio is less than 1 (i.e., |r|<1). If |r|≥1, the sum diverges (goes to infinity or oscillates).

if |r|<1, the sum is given by: \(S_\infty =\frac{a_1}{1-r}\)

Videos

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, …

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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, …

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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.

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Geometric Sequences And Series
A short introduction to geometric sequences and series.

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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, …

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