Sequences

Learning Outcomes

By the end of this section, you will be able to:

  • Find the pattern in a sequence of numbers
  • Write the next terms in a sequence

Inductive Reasoning

Inductive reasoning uses specific examples to draw a generic conclusion.  We will look at patterns of number, or sequences, in this section.  We will use inductive reasoning to look for a pattern in the list of numbers and draw a conclusion on the pattern.  We will then apply that inductive reasoning to the sequence to list the next term (or terms) in the sequence.

Writing the Terms of a Sequence

One way to describe an ordered list of numbers is as a sequence. The sequence established by the number of hits on the website is

[latex]\left\{2,4,8,16,32,\dots \right\}[/latex].

The ellipsis (…) indicates that the sequence continues indefinitely. Each number in the sequence is called a term. The first five terms of this sequence are 2, 4, 8, 16, and 32.

To find the pattern in a sequence, we will look to see if the terms in the sequence are getting smaller or larger.  If the terms are getting smaller, we should check to see if the same number is being subtracted every time or if the same number is being divided each time.  If the terms are getting larger, we should check multiplication and addition between terms.

The sequence {25000, 21600, 18200, 14800, 11400, 8000} gives the value of a truck over six years. As we look at the sequence, we can see that the terms are getting smaller.  So we should check for either subtraction or division. Here, we can see that to get from one term to the next, we subtract 3,400.  So the pattern is to subtract 3,400.  The next number in the sequence would be 4600.

A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.

In this next sequence, you can see the terms are getting larger. So, you should look to see if we are adding the same number each time, or if we are multiplying by the same number each time.  Here, multiplying any term of the sequence by 6 generates the next term.  The next term in the sequence would be 7,776.

A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.

The sequence below is another example where the terms are getting larger. In this case, you can see we are adding 3 each time. You can choose any term of the sequence, and add 3 to find the subsequent term. The next three terms in the sequence would be 18, 21, and 24.

A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.

Examples

Find the pattern in each sequence.

  • {-3, 6, -12, 24, -48, …}
  • {300, 250, 200, 150, 100, …}