Infinite Sequence: Definition & Examples

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  • 0:02 Infinite Concept
  • 0:57 The Nth Term
  • 1:20 Arithmetic / Geometric…
  • 3:13 Other Types of…
  • 3:44 Writing Rules for…
  • 5:50 Lesson Summary
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Lesson Transcript
Instructor: David Liano

David has a Master of Business Administration, a BS in Marketing, and a BA in History.

After completing this lesson, you will be able to identify an infinite sequence and different types of infinite sequences. You will also be able to write additional terms of an infinite sequence and write general rules for infinite sequences.

Infinite Concept

The word 'infinite' implies 'endless' or 'without limit.' Some consider space to be infinite, but we do not really know for sure. Within mathematics, we can apply this concept to the system of natural numbers, using the capital N to represent natural numbers. The ellipsis mark that follows the number 5 tells us that the numbers continue without end.

N = {1, 2, 3, 4, 5, …}

This is an infinite sequence or a sequence with no end. A sequence is a progression of numbers with a clear starting point. The elements of a sequence are not an arbitrary list of numbers. In other words, they are not listed randomly but follow a specific order. Often, infinite sequences follow a specific mathematical pattern so that you can write rules or formulas to easily find any member of the sequence.

The Nth Term

In an infinite sequence, there is a first term, second term, and so on. It is common to represent the nth term of a sequence as a(n). For instance, the first term of a sequence is a(1), and the 23rd term of a sequence is a(23). The numbers in parentheses next to the a are usually written as subscripts.

infinite sequence

Arithmetic Infinite Sequences

An arithmetic infinite sequence is an endless list of numbers in which the difference between consecutive terms is constant. An arithmetic sequence can start at any number, but the difference between consecutive terms, called the common difference, must always be the same.

Let's look at an example of an arithmetic infinite sequence:

5, 8, 11, 14, 17, …

This first term in this sequence is 5, so a(1) = 5. The common difference between consecutive terms is 3.

The difference between consecutive terms can be a negative number:

2, 0.5, -1, -2.5, -4, …

The common difference in this sequence is -1.5.

The set of natural numbers is an arithmetic infinite sequence. It starts at the number 1 and has a common difference of 1.

Geometric Infinite Sequences

In a geometric infinite sequence, there is also a common factor between consecutive terms called the common ratio. The ratio of any term to the previous term is always the same. This ratio holds true for any pair of consecutive terms.

Let's look at an example of a geometric infinite sequence:

2, 4, 8, 16, 32, …

The common ratio in this sequence is 2 because 4/2 = 8/4 = 16/8 = 32/16 = 2. In other words, we multiply any term by the common ratio of 2 to get the next term.

Let's look at another example of a geometric infinite sequence:

27, -9, 3, -1, 1/3, …

The common ratio in this sequence is -1/3.

Other Types of Infinite Sequences

An infinite sequence does not need to be arithmetic or geometric; however, it usually follows some type of rule or pattern. Let's look at this infinite sequence:

1, 4, 9, 16, 25, …

You might notice that this sequence is the square of natural numbers:

12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25

The next term in this sequence would be 62 or 36.

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