1. Defining and Using Sequences and Series
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Use inductive reasoning to find the next terms of a sequence. How can you write it as a function rule?
See solution.
Note that consecutive terms in Sequence (I) have a common difference of 1.5. Conversely, consecutive terms in Sequence (II) have a common ratio of 0.5. To find the rule for the nth 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 |
---|---|---|
a1 | 1 | 1.5(1)−0.5 |
a2 | 2.5 | 1.5(2)−0.5 |
a3 | 4 | 1.5(3)−0.5 |
⋮ | ⋮ | ⋮ |
an | - | 1.5n−0.5 |
This final equation shows the rule for the nth term of Sequence (I). In a similar way, we can find the rule for Sequence II.
Terms | Rule | |
---|---|---|
Sequence I | 1,2.5,4,5.5,7,… | an=1.5n−0.5 |
Sequence II | 8,4,2,1,21,… | bn=8(21)n−1 |