1. Chapter 5: Sequences, Mathematical Induction, and Recursion 5.6 Defining Sequences Recursively 1 So, Nat’ralists observe, a Flea/Hath smaller Fleas on him prey, /And these have smaller Fleas to bite ‘em, /And so proceed ad infinitum. – Jonathan Swift, 1667 – 1745

2. Recurrence Relation A recurrence relation for a sequence a 0, a 1, a 2, … is a formula that relates each term a k, to certain of its predecessors a k-1, a k-2, …, a k-i where i is an integer with k – i  0. Initial Conditions The initial conditions for such a recurrence relation specify the values of a 0, a 1, a 2, …, a i-1, if i is a fixed integer, or a 0, a 1, a 2, …, a m where m is an integer with m  0, if i depends on k. 5.6 Defining Sequences Recursively2

3. Find the first four terms of the recursively defined sequence. 5.6 Defining Sequences Recursively3

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5. Fibonacci proposed the following problem: A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions: 1.Rabbit pairs are not fertile during their first month of life but thereafter give birth to one new male/female pair at the end of every month. 2.No rabbits die. How many rabbits will there be at the end of the year? 5.6 Defining Sequences Recursively5

6. The solution is a recurrence relation 5.6 Defining Sequences Recursively6

7. Please read this section in your textbook. 5.6 Defining Sequences Recursively7

8. F 0, F 1, F 2, … is the Fibonacci sequence. 5.6 Defining Sequences Recursively8

9. Given numbers a 1, a 2, …, a n, where n is a positive integer, the summation from i = 1 to n of the a i is defined as follows: if n > 1. the product from i = 1 to n of the a i is defined by: if n > 1. 5.6 Defining Sequences Recursively9

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