## Table of Contents

- What Is a Geometric Sequence?
- Geometric Sequence Formula
- How to Find a Term in a Geometric Sequence?
- Examples of Geometric Sequence
- Lesson Summary

Get the geometric sequence definition and view examples. Learn how to find the nth term of a geometric sequence using the geometric sequence formula.
Updated: 11/12/2021

- What Is a Geometric Sequence?
- Geometric Sequence Formula
- How to Find a Term in a Geometric Sequence?
- Examples of Geometric Sequence
- Lesson Summary

A **geometric sequence** is defined as "a sequence (that is, a set of ordered elements) where the ratio between two consecutive **terms** is always the same number, known as the constant ratio." In other words, to go from one term to the next, multiply the former by the ratio.

Now that the question "what is a geometric sequence?" was addressed, consider the following example: {eq}2,~6,~18,~54,~\cdots {/eq}. The suspension points after the fourth element indicate that the sequence continues indefinitely, so it is an infinite geometric sequence. In the given example, the constant ratio is {eq}3 {/eq}. The elements of a geometric sequence are said to be its terms and one might be interested in finding a generic term, say, the n-th term {eq}a_n {/eq}, given the initial term ({eq}a_1 {/eq}) and the **common ratio** {eq}r {/eq}. In this case, the general term is given by {eq}a_n~=~a_1r^{n-1} {/eq}, often referred to as the geometric sequence equation.

Geometric sequences can be classified according to the number of terms that they have. A finite geometric sequence is one that contains a finite number of terms.

An infinite geometric sequence, in turn, is one that has infinite terms; that is, there is no last term in this type of sequence. Their representation usually ends with suspension points to indicate that the terms continue following the pattern of the terms displayed.

In the first section of this lesson, the geometric sequence formula, which tells how to find the nth term of the geometric sequence, was provided. Here is an explanation as to why that formula gives a general term of the geometric sequence.

Consider the first few elements of a geometric sequence, which are going to be denoted by {eq}a_i {/eq}, where {eq}I {/eq} is an index associated with the position of the term. Geometric sequences have a special property that, to find the subsequent term, one multiplies the previous one by the common ratio, which is going to be denoted here by {eq}r {/eq}.

$$a_1~=~a_1\\ a_2~=~a_1r\\ a_3~=~a_2r~=~(a_1r)r~=~a_1r^2\\ a_4~=~a_3r~=~(a_1r^2)r~=~a_1r^3\\ a_5~=~a_4r~=~(a_1r^3)r~=~a_1r^4\\ \vdots $$

Therefore, proceeding with analogous reasoning, one might conclude that the geometric sequence formula for the nth term is {eq}a_n=a_1r^{n-1} {/eq}, where {eq}a_1 {/eq} is the first term, {eq}r {/eq} is the common ratio, and {eq}n {/eq} is the number of terms.

In a problem involving geometric sequences, some of the elements will be given and others will be desired. One of the elements that might be wanted is the common ratio. How to find {eq}r {/eq} in a geometric sequence? Simply divide any term (except for the first one) by its predecessor to obtain {eq}r {/eq}. In the first example, sequence {eq}2,~6,~18,~54,~\cdots {/eq} was given and it was claimed that the common ratio was {eq}3 {/eq}. Here is why: {eq}\frac{a_2}{a_1}~=~\frac{6}{2}~=~3 {/eq}. If several elements of a sequence are given, depending on the pair that is chosen to evaluate {eq}r {/eq}, the calculations can be easier, so one should choose the terms with that in mind.

As was explained previously, geometric sequences can be classified into finite and infinite. For the finite ones, the sum of their terms is given by a formula that will be presented here. The sum of the terms of a geometric sequence is referred to as a geometric series, which is finite or infinite depending on the number of elements involved.

Let {eq}S {/eq} denote the sum of the elements of a finite geometric sequence. Then, {eq}S~=~a_1~+~a_2~+~\cdots~+~a_n {/eq}. Multiplying the equation by the common ratio {eq}q {/eq} on both sides gives {eq}qS~=~qa_1~+~qa_2~+~qa_3~+~\cdots~+~qa_n~=~a_2~+~a_3~+~a_4~+~\cdots~+~a_1q^n {/eq}. Subtracting the second equation from the first yields $$S~-~qS~=~a_1~-~a_1q^n~\implies~S~=~a_1\frac{1 - q^n}{1-q} $$

Given the formula for the **general term** of a geometric sequence, a coup;e of questions that arise are: how to find a term in a geometric sequence? What about the number of terms? Of course, in this type of situation, some pieces of information like the common ratio and the first term must be provided.

To find the fifth term, for example, given the ratio {eq}q {/eq} and the first term {eq}a_1 {/eq}, simply plug those numbers in the formula along with {eq}n~=~5 {/eq}, obtaining {eq}a_5~=~a_1q^4 {/eq}.

Now, to determine the number of elements in a finite sequence, given the first and last terms as well as the ratio, simply isolate the term with exponent {eq}n~-~1 {/eq} and solve an exponential equation for {eq}n {/eq}.

To bring the abstract types of problems from the previous section to a more tangible context, consider the following questions.

- Example 1

Find the fifth element of the infinite geometric sequence that has {eq}2 {/eq} as the first element and common ratio.

It is given in the problem that {eq}a_1~=~2 {/eq} and {eq}q~=~2 {/eq}. Plugging those values in the general term formula, along with {eq}n~=~5 {/eq} gives {eq}a_5~=~(2)2^4~=~32 {/eq}. Plugging the values in a graph, one can observe that an exponential function restricted to natural numbers is the model for a geometric sequence, as shown in Figure 1.

- Example 2

The first and last terms of a geometric sequence are, respectively {eq}1 {/eq} and {eq}\frac{1}{81} {/eq}. Determine the number of elements of the sequence knowing that the common ratio is {eq}\frac{1}{3} {/eq}.

Plugging the values in the formula yields {eq}\frac{1}{81}~=~1\left(\frac{1}{3}\right)^{n-1}~\implies~\left(\frac{1}{3}\right)^4~=~\left(\frac{1}{3}\right)^{n-1}~\implies~n-1~=~4~\implies~n~=~5 {/eq}. There are five elements in the given geometric sequence and its geometric interpretation is depicted in Figure 2.

A **geometric sequence** (also called geometric progression) is one where the ratio between two consecutive elements, called **terms**, is the same number. For this reason, it is called the **common ratio**. Geometric sequences can be finite or infinite depending on the number of terms that it has. The **general term** of a geometric progression is {eq}a_n~=~a_1q^{n-1} {/eq}, with {eq}a_1 {/eq} being the first term, {eq}q {/eq}, the common ratio, and {eq}n {/eq} indicating the position of the desired term.

The geometric interpretation for this type of progression is an exponential function restricted to natural numbers. If the sequence is set to begin at {eq}a_0 {/eq}, instead of {eq}a_1 {/eq}, then the point {eq}(0,~a_0) {/eq} is included in the graph of the geometric interpretation.

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Frequently Asked Questions

To have a geometric sequence we need an initial term a1 and a common ratio q. The general formula for the nth term of this sequence is an = a1q^(n-1).

Let a1 be the first term of a geometric series, q the common ratio, and n the nth term. The general term formula is an = a1q^(n-1).

One way of writing a geometric sequence is listing its terms a1, a2, ..., an if it is a finite sequence or a1, a2, a3, ... if it is an infinite one.

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