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Arithmetic Sequences

Writing Recursive Rules for Arithmetic Sequences

There are different ways to express the rules of sequences depending on the information that's known. Explicit rules require the first term, the common difference, and the position of the desired term Recursive rules have different requirements.

Concept

Recursive Rule

A recursive rule gives the first term or terms of a sequence and describes how each term is related to the preceding term(s) with a recursive equation. For example, arithmetic and geometric sequences can be described recursively.

Example arithmetic sequence Example geometric sequence
geometric sequence
geometric sequence
Recursive equation
One particularly well-known sequence that is defined recursively is the Fibonacci sequence, in which each term is the sum of the two previous terms.

Method

Writing a Recursive Rule for an Arithmetic Sequence

The recursive rule for an arithmetic sequence can be expressed as where and are consecutive terms and is the common difference. To write the rule, must be found. Consider the arithmetic sequence To find we can subtract any term from the term that comes after it. Let's use and Using the recursive rule can be written as As it's written, this rule describes any sequence with a common difference of For example, it can describe as well as To ensure the recursive rule defines the given sequence, it is necessary to also give the first term, Thus, the recursive rule for the given sequence is Now that the recursive rule is known, it can be used to find any term, provided that the previous term is given. For example, since can be found.

Thus, the term in the sequence is
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Exercise

Bonnie recently purchased a used car. Each weekend Bonnie records the odometer of her car. The first four weekends she noted the following miles.

Create a recursive rule that describes the sequence. Then use the recursive rule to find the odometer's reading the following two weeks, assuming that Bonnie drives exactly the same distance every week.

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Solution

Before we write the recursive rule for the sequence, let's ensure it's arithmetic.

Notice that Bonnie drives miles each week. Therefore, the odometer readings form an arithmetic sequence with a common difference of Using and the given information we can write the recursive rule as follows. We can use the recursive rule to find the odometer readings for the next two weeks, and We'll use to find Similarly, we can add to to find the odometer reading of the next weekend. Thus, after the fifth weekend, the odometer will read miles and after the sixth weekend, it will read miles.

Method

Translating between Explicit and Recursive Rules

As has been explored, it is possible to express sequences with explicit rules and recursive rules. Since the rules present different information about the sequence, it can be useful to translate between the two.
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Exercise

An arithmetic sequence is given by the following explicit rule. Translate it into a recursive rule.

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Solution

A recursive rule for an arithmetic sequence is written as

where is the first term and is the common difference. Since an explicit rule for an arithmetic sequence is written as it can be seen that and Thus, we can write the recursive rule for this sequence as

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Exercise

An arithmetic sequence is defined by the following recursive rule. Write the explicit rule for the same sequence.

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Solution

The explicit rule of an arithmetic sequence is written as where is the first term and is the common difference. To write an explicit rule, both and are needed. From the recursive rule it can be seen that and Thus, the explicit rule can be written and simplified as follows.

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