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Recursive Formulas

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

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

In the above sequence, a_1=4 , a_2=7 , a_3=11 and a_4=16 .

Example A

Describe the pattern and write a recursive rule for the sequence: 9, 11, 13, 15, \ldots

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 a_{n-1} and a_n to write a recursive rule as follows: a_n=a_{n-1}+29

Example B

Write a recursive rule for the sequence: 3, 9, 27, 81, \ldots

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

Example C

Write a recursive rule for the sequence: 1, 1, 2, 3, 5, 8, \ldots

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: a_n=a_{n-2}+a_{n-1} .

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 a_{n-1} and a_n to write a recursive rule as follows: a_n=a_{n-1}+29

Guided Practice

Write the recursive rules for the following sequences.

1. 1, 2, 4, 8, \ldots

2. 1, -2, -5, -8, \ldots

3. 1, 2, 4, 7, \ldots

Answers

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

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

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

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)

Vocabulary

Recursive Rule
A rule that can be used to calculate a term in a sequence given the previous term(s).

Practice

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

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

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