Geometric Sequence Formula and Examples

Anderson Gomes Da Silva, Yuanxin (Amy) Yang Alcocer
  • Author
    Anderson Gomes Da Silva

    Anderson holds a Bachelor's and Master's Degrees (both in Mathematics) from the Fluminense Federal University and the Pontifical Catholic University of Rio de Janeiro, respectively. He was a Teaching Assistant at the University of Delaware (UD) for two and a half years, leading discussion and laboratory sessions of Calculus I, II and III. In the Winter of 2021 he was the sole instructor for one of the Calculus I sections at UD.

  • Instructor
    Yuanxin (Amy) Yang Alcocer

    Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

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

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What Is a Geometric Sequence?

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.

Finite Geometric Sequence

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.

Infinite Geometric Sequence

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.

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  • 0:01 A Geometric Sequence
  • 1:37 The General Term
  • 2:23 Why We Use It
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Geometric Sequence Formula

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.

How to find r in a Geometric Sequence?

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.

Geometric Sequence Sum Formula

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 $$S~-~qS~=~a_1~-~a_1q^n~\implies~S~=~a_1\frac{1 - q^n}{1-q} $$

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Frequently Asked Questions

What is the general formula for a geometric sequence?

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).

How do you find the nth term in a geometric sequence?

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).

How do you write a geometric sequence?

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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