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Recursive Formulas

Where a term is based on prior term(s) in a sequence.

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Describing the Pattern and Writing a Recursive Rule for a Sequence

In 2013, the days a full moon appeared were in the following sequence (with Jan. 1 being Day 1). Write a recursive formula for the sequence.

9, 38, 67, 96, ...

Source: Moongiant

Recursive Rule

A recursive rule for a sequence is a formula which tells us how to progress from one term to the next in a sequence. Generally, the variable n\begin{align*}n\end{align*} is used to represent the term number. In other words, n\begin{align*}n\end{align*} takes on the values 1 (first term), 2 (second term), 3 (third term), etc. The variable, an\begin{align*}a_n\end{align*} represents the nth\begin{align*}n^{th}\end{align*} term and an1\begin{align*}a_{n-1}\end{align*} represents the term preceding an\begin{align*}a_n\end{align*}.

Example sequence: 4,7,11,16,,an1,an\begin{align*}4, 7, 11, 16, \ldots, a_{n-1}, a_n\end{align*}

In the above sequence, a1=4\begin{align*}a_1=4\end{align*}, a2=7\begin{align*}a_2=7\end{align*}, a3=11\begin{align*}a_3=11\end{align*} and a4=16\begin{align*}a_4=16\end{align*}.

Let's describe the pattern and write a recursive rule for the sequence: 9,11,13,15,\begin{align*}9, 11, 13, 15, \ldots\end{align*}

First we need to determine what the pattern is in the sequence. If we subtract each term from the one following it, we see that there is a common difference of 29. We can therefore use an1\begin{align*}a_{n-1}\end{align*} and an\begin{align*}a_n\end{align*} to write a recursive rule as follows: an=an1+29\begin{align*}a_n=a_{n-1}+29\end{align*}

Now, let's write a recursive rule for the following sequences.

1. 3,9,27,81,\begin{align*}3, 9, 27, 81, \ldots\end{align*}

In this sequence, each term is multiplied by 3 to get the next term. We can write a recursive rule: an=3an1\begin{align*}a_n=3a_{n-1}\end{align*}

1. 1,1,2,3,5,8,\begin{align*}1, 1, 2, 3, 5, 8, \ldots\end{align*}

This is a special sequence called the Fibonacci sequence. In this sequence each term is the sum of the previous two terms. We can write the recursive rule for this sequence as follows: an=an2+an1\begin{align*}a_n=a_{n-2}+a_{n-1}\end{align*}.

Examples

Example 1

Earlier, you were asked to write a recursive formula for the sequence 9, 38, 67, 96, ...

First we need to determine what pattern the sequence is following. If we subtract each term from the one following it, we find that there is a common difference of 29. We can therefore use an1\begin{align*}a_{n-1}\end{align*} and an\begin{align*}a_n\end{align*} to write a recursive rule as follows: an=an1+29\begin{align*}a_n=a_{n-1}+29\end{align*}

Write the recursive rules for the following sequences.

Example 2

1,2,4,8,\begin{align*}1, 2, 4, 8, \ldots\end{align*}

In this sequence each term is double the previous term so the recursive rule is: \begin{align*}a_n=2a_{n-1}\end{align*}

Example 3

\begin{align*}1, -2, -5, -8, \ldots\end{align*}

This time three is subtracted each time to get the next term: \begin{align*}a_n=a_{n-1}-3\end{align*}.

Example 4

\begin{align*}1, 2, 4, 7, \ldots\end{align*}

This one is a little trickier to express. Try looking at each term as shown below:

\begin{align*}a_1 &=1 \\ a_2 &=a_1+1 \\ a_3 &=a_2+2 \\ a_4 &=a_3+3 \\ & \ \ \vdots \\ a_n &=a_{n-1}+(n-1)\end{align*}

Review

Describe the pattern and write a recursive rule for the following sequences.

1. \begin{align*}\frac{1}{4}, -\frac{1}{2}, 1, -2 \ldots\end{align*}
2. \begin{align*}5, 11, 17, 23, \ldots\end{align*}
3. \begin{align*}33, 28, 23, 18, \ldots\end{align*}
4. \begin{align*}1, 4, 16, 64, \ldots\end{align*}
5. \begin{align*}21, 30, 39, 48, \ldots\end{align*}
6. \begin{align*}100, 75, 50, 25, \ldots\end{align*}
7. \begin{align*}243, 162, 108, 72, \ldots\end{align*}
8. \begin{align*}128, 96, 72, 54, \ldots\end{align*}
9. \begin{align*}1, 5, 10, 16, 23, \ldots\end{align*}
10. \begin{align*}0, 2, 2, 4, 6, \ldots\end{align*}
11. \begin{align*}3, 5, 8, 12, \ldots\end{align*}
12. \begin{align*}0, 2, 6, 12, \ldots\end{align*}
13. \begin{align*}4, 9, 14, 19, \ldots\end{align*}
14. \begin{align*}\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5} \ldots\end{align*}
15. \begin{align*}4, 5, 9, 14, 23, \ldots\end{align*}

To see the Review answers, open this PDF file and look for section 11.2.

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
common difference Every arithmetic sequence has a common or constant difference between consecutive terms. For example: In the sequence 5, 8, 11, 14..., the common difference is "3".
common ratio Every geometric sequence has a common ratio, or a constant ratio between consecutive terms. For example in the sequence 2, 6, 18, 54..., the common ratio is 3.
index The index of a term in a sequence is the term’s “place” in the sequence.
recursive The recursive formula for a sequence allows you to find the value of the nth term in the sequence if you know the value of the (n-1)th term in the sequence.
recursive formula The recursive formula for a sequence allows you to find the value of the nth term in the sequence if you know the value of the (n-1)th term in the sequence.
sequence A sequence is an ordered list of numbers or objects.