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High School Geometry: Tutoring Solution14 chapters | 161 lessons

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Lesson Transcript

Instructor:
*David Liano*

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.

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, â€¦}

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.

In an infinite sequence, there is a first term, second term, and so on. It is common to represent the *n*th 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.

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, â€¦

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, â€¦

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.

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, â€¦

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, â€¦

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

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, â€¦

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.

Writing a rule for the nth term of an infinite sequence can be tricky. A member of an infinite sequence is usually a function of its order in the sequence. Let's consider this sequence:

2, 4, 6, 8, 10, â€¦

*a*(1) = 2, *a*(2) = 4, *a*(3) = 6, etc.

Did you notice that the value of each term is double its order in the sequence? Therefore, the general rule for this infinite sequence is *a*(*n*) = 2*n*.

Let's try one that is more difficult. We need to write a general rule for the *n*th term of the following infinite sequence:

0, 7, 26, 63, 124, â€¦

We need to find a rule that gives us 0 when *n* = 1; 7 when *n* = 2; 26 when *n* = 3; and so on. Each of these terms is very close to the cube of its order in the sequence. In fact, each term is always one less than the cube of its order. For example, *a*(3) = 3*a*(3 - 1). Therefore, the general rule for this sequence is *a*(*n*) = *n**a*(3 - 1).

Let's try one more example. We need to write the next term of the following infinite sequence and then write a general rule for the *n*th term.

1, 8, 64, 512, 4,096, â€¦

These terms seem to be powers of the number 8. Therefore, the next term, *a*(6), equals (8)4,096 = 32,768.

Our first try at a general rule might be *a*(*n*) = 8*n*. But that does not work because that would give us a first term of *a*`(1)` = 81 = 8. It looks like the order of each power is out of place by one. But we can fix this by adjusting our general rule to be *a*(*n*) = 8(*n* - 1). Now it works.

An **infinite sequence** is a list of terms that continues forever. The terms are ordered. This means that the first member is always the first member, and the 15th member is always the 15th member. This lesson focused on infinite sequences that contained some type of mathematical pattern. In an **arithmetic infinite sequence**, the difference between consecutive terms, called the common difference, must always be the same. A **geometric infinite sequence** also has a common factor between consecutive terms called the common ratio. The ratio of any term to the previous term is always the same. An infinite sequence does not need to be arithmetic or geometric, but it usually follows some type of rule or pattern.

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High School Geometry: Tutoring Solution14 chapters | 161 lessons

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