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Recurrence Relations in Discrete Structures

Explore the concept of recurrence relations for sequences, solving counting problems, modeling with examples, and applications like the Fibonacci sequence. Includes exercises for practice.

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Recurrence Relations in Discrete Structures

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  1. Recurrence Relations Section 6.1 CSE 2813 Discrete Structures

  2. Definition • A recurrence relation for the sequence {an} is an equation that expresses an in terms of one or more of the previous terms of the sequence, namely, a0, a1,…,an-1, for all integers n with n n0, where n0 is a nonnegative integer. • A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. CSE 2813 Discrete Structures

  3. Recurrence Relations vs. Recursive Definitions • So what is the difference? • Recursive definitions can be used to solve counting problems. When they are used in this way, the rule for finding terms from those that precede them is called a recurrence relation. CSE 2813 Discrete Structures

  4. Example • Let {an} be a sequence that satisfies the recurrence relation an= an-1  an-2 for n =2, 3, 4,… Suppose that a0=3 and a1=5. • What are a2 and a3? CSE 2813 Discrete Structures

  5. Example • Consider the recurrence relation: an = 2an-1 an-2 for n =2, 3, 4, … • Show whether each of the following is a solution of this recurrence relation? an=3n an=2n an=5 CSE 2813 Discrete Structures

  6. Modeling with Recurrence Relations • A person deposits $10,000 in a savings account at a bank yielding 11% per year with interest compounded annually. • How much will be in the account after 30 years? CSE 2813 Discrete Structures

  7. Rabbits and theFibonacci Sequence • A young pair of rabbits (one of each sex) is placed on an island. • A pair does not breed until they are 2 months old. • After they are 2 months old, each pair produces another pair each month. • Find the number of pairs of rabbits on the island after n months, assuming that no rabbits ever die. CSE 2813 Discrete Structures

  8. The Tower of Hanoi • Find a recurrence relation to find the number of moves needed to solve the Tower of Hanoi problem with n disks. Tower of Hanoi CSE 2813 Discrete Structures

  9. More Example • Find a recurrence relation for the number of bit strings of length n that do not contain two consecutive 0s. • Find a recurrence relation for the number of bit strings of length n that contain two consecutive 0s. CSE 2813 Discrete Structures

  10. Exercises • 1, 5, 9, 10, 11, 13, 24, 25, 27 CSE 2813 Discrete Structures

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