Big Ideas Math Algebra 2, Virginia Edition, 2019
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Big Ideas Math Algebra 2, Virginia Edition, 2019 View details
1. Defining and Using Sequences and Series
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Exercise 2 Page 389

Use inductive reasoning to find the next terms of a sequence. How can you write it as a function rule?

See solution.

Practice makes perfect
When we are given a sequence, generally we see its first few terms.
To find the next terms we can reason inductively. To do this can follow several strategies, like finding a relationship between consecutive terms. Let's examine two examples.

Note that consecutive terms in Sequence (I) have a common difference of Conversely, consecutive terms in Sequence (II) have a common ratio of To find the rule for the term, we can rewrite each term in terms of the common difference or the common ratio. Let's do it for Sequence (I).

Term Value In Terms of the Common Difference
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This final equation shows the rule for the term of Sequence (I). In a similar way, we can find the rule for Sequence II.

Terms Rule
Sequence I
Sequence II
However, there are sequences in which the relationship between the terms cannot be explained in the same way. When this is the case we look for other recognizable patterns. Let's investigate another sequence.
Note that the denominator of all the terms in the above sequence is Moreover, the numerator is the square of the term's position in the sequence. Considering this, we can write a rule for the term.
As a result, to write the rule for the term of a sequence we use inductive reasoning to recognize any visible pattern. Then, we use this pattern to write the rule.