Chapter 7: Recurrence Relations

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Chapter 7 Recurrence Relations

• Discrete Mathematics
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Recurrence Relation

• A recurrence relation for the sequence a0,
• a1,is an equation that relates an to certain
• of its predecessors a0, a1,, an-1.
• Initial conditions for the sequence a0, a1,
• are explicitly given values for a finite number
• of the terms of the sequence.

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Example 7.1.1

• The Fibonacci sequence is defined by the
• recurrence relation
• and initial conditions

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Example 7.1.2

• A person invests 1,000 at 12 percent
• interest compounded annually. If An
• represents the amount at the end of n years,
• find a recurrence relation and initial
• conditions that define the sequence An.

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Solving Recurrence Relations

• To solve a recurrence relation involving the
• sequence a0, a1, is to find an explicit
• formula for the general term an.
• There are two methods of solving recurrence
• relations
• Iteration
• Linear homogeneous recurrence relations with
  constant coefficients.

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Solving Recurrence Relations by Iteration

• To solve a recurrence relation involving the
• sequence a0, a1, by iteration, we use the
• recurrence relation to write the nth term an
• In terms of certain of its predecessors
• an-1,,a0.
• We then successively use the recurrence
• relation to replace each of an-1, by certain
• of their predecessors.

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Example 7.2.1

• Solving the recurrence relation
• subject to the initial condition
• by iteration.

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Example 7.2.2

• Solving the recurrence relation Sn 2Sn-1
• subject to the initial condition S0 1 by
• iteration.

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Definition 7.2.6

• A linear homogeneous recurrence relation of order
  
• k with constant coefficients is a recurrence
  relation
• of the form
• Notice that a linear homogeneous recurrence
• relation of order k with constant coefficients,
• together with the k initial conditions
• uniquely defines a sequence a0, a1,.

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Theorem 7.2.11

• Let be a
  second-order,
• linear homogeneous recurrent relation with
• constant coefficients.
• If S and T are solutions of
• then U bS dT is also a solution of

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Example

• Solving recurrence relation
• and initial conditions

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Example 7.2.13

• Find an explicit formula for the Fibonacci
• sequence.

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Theorem 7.2.14

• Let be a
  second-order,
• linear homogeneous recurrent relation with
• constant coefficients and initial conditions
• a0 C0, a1 C1.
• If both roots of
• are equal to r, then there exist constants b
• and d such that

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Example 7.2.15

• Solve the recurrence relation
• subject to the initial conditions