11.5 Recursive Rules for Sequences

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11.5 Recursive Rules for Sequences

• p. 681

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• What is a recursive rule for sequences?
• What does ! mean in math?

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Explicit Rule

• A function based on a terms position, n, in a
  sequence.
• All the rules for the nth term that weve been
  working with are explicit rules such as
  ana1rn-1.

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Recursive Rule

• Gives the first term(s) of a sequence and an
  equation that relates the given term(s) to the
  next terms in the sequence.
• For example Given a01 and anan-1-2
• The 1st five terms of this sequence would be a0,
  a1, a2, a3, a4 OR
• 1, -1, -3, -5, -7

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Example Write the 1st 5 terms of the sequence.

• a12, a22, anan-2-an-1
• a3a3-2-a3-1?a1-a22-20
• a4a4-2-a4-1?a2-a32-02
• a5a5-2-a5-1?a3-a40-2-2
• 2, 2, 0, 2, -2

2nd term
1st term
1 2 3 4 5
2 2 0 2 -2
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Example Write the indicated rule for the
arithmetic sequence with a115 and d5.

• Recursive rule
• (Use the idea that you get the next term by
  adding 5 to the previous term.)
• Or anan-15
• So, a recursive rule would be a115, anan-15

• Explicit rule
• ana1(n-1)d
• an15(n-1)5
• an155n-5
• an105n

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Example Write the indicated rule for the
geometric sequence with a14 and r0.2.

• Explicit rule
• ana1rn-1
• an4(0.2)n-1

• Recursive rule
• (Use the idea that you get the next term by
  multiplying the previous term by 0.2)
• Or anran-10.2an-1
• So, a recursive rule for the sequence would be
  a14, an0.2an-1

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• What is a recursive rule for sequences?
• Gives the first term(s) of a sequence and an
  equation that relates the given term(s) to the
  next terms in the sequence.

• What does ! mean in math?
• Factorial 4! 4321

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Assignment

• p. 684
• 14-25odd
• 27-33 odd

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Write a recursive rule for the sequence
1,2,2,4,8,32, .

• First, notice the sequence is neither arithmetic
  nor geometric.
• So, try to find the pattern.
• Notice each term is the product of the previous 2
  terms.
• Or, an-1an-2
• So, a recursive rule would be
• a11, a22, an an-1an-2

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Example Write a recursive rule for the sequence
1,1,4,10,28,76.

• Is the sequence arithmetic, geometric, or
  neither?
• Find the pattern.
• 2 times the sum of the previous 2 terms
• Or 2(an-1an-2)
• So the recursive rule would be
• a11, a21, an 2(an-1an-2)