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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 was in the following sequence (with Jan. 1 being Day 1). Write a recursive formula for the sequence.

9, 38, 67, 96, ...

[Source: http://www.moongiant.com/Full_Moon_New_Moon_Calendar.php]

Guidance

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 \begin{align*}n\end{align*} is used to represent the term number. In other words, \begin{align*}n\end{align*} takes on the values 1 (first term), 2 (second term), 3 (third term), etc. The variable, \begin{align*}a_n\end{align*} represents the \begin{align*}n^{th}\end{align*} term and \begin{align*}a_{n-1}\end{align*} represents the term preceding \begin{align*}a_n\end{align*}.

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

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

Example A

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

Solution: 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 \begin{align*}a_{n-1}\end{align*} and \begin{align*}a_n\end{align*} to write a recursive rule as follows: \begin{align*}a_n=a_{n-1}+29\end{align*}

Example B

Write a recursive rule for the sequence: \begin{align*}3, 9, 27, 81, \ldots\end{align*}

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

Example C

Write a recursive rule for the sequence: \begin{align*}1, 1, 2, 3, 5, 8, \ldots\end{align*}

Solution: 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: \begin{align*}a_n=a_{n-2}+a_{n-1}\end{align*}.

Intro Problem Revisit 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 \begin{align*}a_{n-1}\end{align*} and \begin{align*}a_n\end{align*} to write a recursive rule as follows: \begin{align*}a_n=a_{n-1}+29\end{align*}

Guided Practice

Write the recursive rules for the following sequences.

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

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

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

Answers

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

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

3. 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*}

Explore More

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*}

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 11.2. 

Vocabulary

common difference

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

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

index

The index of a term in a sequence is the term’s “place” in the sequence.
recursive

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

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

sequence

A sequence is an ordered list of numbers or objects.

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