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

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

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

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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

The solution is a recurrence relation 5.6 Defining Sequences Recursively6

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

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

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 > Defining Sequences Recursively9

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