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Introduction to Geometric Sequences

Author: Sophia

what's covered
In this lesson, you will learn how to determine the formula for a given geometric sequence. Specifically, this lesson will cover:

Table of Contents

1. Geometric Sequence Defined

A sequence is a set of numbers in numerical order. You may be familiar with arithmetic sequences, which have a common difference between each term, a constant number that is either added or subtracted as we go from one term to the next.

EXAMPLE

Consider the arithmetic sequence left curly bracket 3 comma space 5 comma space 7 comma space 9... right curly bracket, with a common ratio of 2, where we add 2 to get the value of the next term: 3 plus 2 is 5, 5 plus 2 is 7, 7 plus 2 is 9, etc.

With geometric sequences, there is a common ratio between each term. Similar to arithmetic sequences, the common ratio is a constant value; however it is not added or subtracted from one term to the next, it is multiplied by the preceding term to continue the sequence.

EXAMPLE

Consider the geometric sequence left curly bracket 3 comma space 6 comma space 12 comma space 24 comma space 48... right curly bracket, with a common ratio of 2. We can multiply a term by 2 to get the value of the next term: 3 times 2 is 6, 6 times 2 is 12, 12 times 2 is 48, etc. The next term in the sequence would be 96, and 192 would follow that.

terms to know
Geometric Sequence
A set of numbers in numerical order, with a non-zero common ratio between each term.
Common Ratio
The ratio between any two consecutive terms in a geometric sequence, a constant value.


2. Divergent and Convergent Sequences

The previous sequence was also an example of a divergent sequence. A divergent sequence is a sequence whose terms do not have a finite limit; they tend toward positive or negative infinity.

Since we are continuously multiplying each term in the sequence by 2 in the above example, we know that the terms will continue to increase in value, making the limit of values positive infinity. Even if the common ratio were -2, the terms would alternate between positive and negative, but would still grow in absolute magnitude. Divergent sequences are characterized by a common ratio greater than 1 or less than -1.

EXAMPLE

Consider if we have a geometric sequence with a common ratio of 5. The terms will tend toward positive infinity as we continue:

open curly brackets 5 comma space 25 comma space 125 comma space 625... close curly brackets

In contrast, some geometric sequences are convergent. The terms in convergent sequences do have a limit. Convergent sequences are characterized by a common ratio between -1 and 1.

EXAMPLE

Consider if we have a geometric sequence with a common ratio of 0.5, as in the example below:

left curly bracket 768 comma space 384 comma space 192 comma space 96... right curly bracket

In this sequence, each term is multiplied by 0.5, or in other words, cut in half, as the sequence continues. Eventually, the term would be virtually zero, thus the sequence is convergent. Even if the common ratio were -0.5, the terms would still converge to zero, even though they would alternate between positive and negative.

big idea
Divergent and convergent sequences are characterized by their common ratio, r:
  • If open vertical bar r close vertical bar greater than 1, the sequence is divergent.
  • If open vertical bar r close vertical bar less than 1, the sequence is convergent.

terms to know
Divergent Sequence
A sequence whose terms do not have a finite limit; they tend toward positive or negative infinity.
Convergent Sequence
A sequences whose terms have a finite limit; they tend toward a specific value.


3. Formula for Geometric Sequences

Let's return to the geometric sequence: left curly bracket 3 comma space 6 comma space 12 comma space 24 comma space 48... right curly bracket. It was easy enough to find the value of the next two terms, by multiplying 48 by 2 to get 96, and then by 2 again to get 192. What if we wanted to find the value of the 78th term? Or even the 115th therm? Certainly finding the value of that term by continuously multiplying by 2 is inefficient.

To find the nth term of a sequence, we can use the geometric sequence formula:

formula to know
Geometric Sequence
a subscript n equals a subscript 1 times r to the power of n minus 1 end exponent

In this formula:

  • a subscript n is the value of the nth term.
  • a subscript 1 is the value of the first term.
  • r is the common ratio.
  • n is the term number.

4. Writing a Formula for a Geometric Sequence

EXAMPLE

Consider the geometric sequence left curly bracket 8748 comma space 2916 comma space 972 comma space 324... right curly bracket.

How can we develop a formula to represent this sequence, so that we may use it to find the value of any term? We will need to identity a subscript 1 and r:

  • a subscript 1: The value of the first term is 8748, so this will be a subscript 1 in our formula.
  • r: We need to calculate the common ratio, r. To do so, we will take any two consecutive terms, and divide the second by the first, for instance, 324 divided by 972 equals 1 third. This tells us that each term is multiplied by 1 third to get the value of the next term.
Plugging in 8748 for a subscript 1 and 1 third for r, the formula for this sequence is:

a subscript n equals 8748 times open parentheses 1 third close parentheses to the power of n minus 1 end exponent


5. Using the Formula to find a Term

EXAMPLE

Use the above formula a subscript n equals 8748 times open parentheses 1 third close parentheses to the power of n minus 1 end exponent to find the value of the 7th term in the sequence.

Since we want to find the 7th term, we use plug 7 for n in our formula:

a subscript n equals 8748 times open parentheses 1 third close parentheses to the power of n minus 1 end exponent Plug in 7 for n
a subscript n equals 8748 times open parentheses 1 third close parentheses to the power of 7 minus 1 end exponent Evaluate the exponent
a subscript n equals 8748 times open parentheses 1 third close parentheses to the power of 6 Raise open parentheses 1 third close parentheses to the 6th power
a subscript n equals 8748 times open parentheses 1 to the power of 6 over 3 to the power of 6 close parentheses Evaluate fraction
a subscript n equals 8748 times open parentheses 1 over 729 close parentheses Multiply 8748 and 1 over 729
a subscript n equals 8748 over 729 Divide 8749 by 729
a subscript n equals 12 Our solution

summary
A geometric sequence is a set of numbers in numerical order with a nonzero common ratio between each term. There are two types of sequences: divergent and convergent sequences. Divergent sequences have a common ratio, r, such that the absolute value of r is greater than 1. Convergent sequences have a common ratio such that the absolute value of r is less than 1. The formula for geometric sequences is a sub n equals a sub 1 times r to the n minus one power. When writing a formula for a geometric sequence, you need to determine the common ratio by dividing any term in the sequence by the term before it. You can also use the formula to find any term in the sequence.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
Common Ratio

The ratio between any two consecutive terms in a geometric sequence; a constant value.

Convergent Sequence

A sequence whose terms have a finite limit; they tend toward a specific value.

Divergent Sequence

A sequence whose terms do not have a finite limit; they tend toward ± infinity.

Geometric Sequence

A set of numbers in numerical order, with a non-zero common ratio between each term.

Formulas to Know
Geometric Sequence

a subscript n equals a subscript 1 times r to the power of left parenthesis n minus 1 right parenthesis end exponent