chapter 5: sequences, mathematical induction, and recursion 5.6 defining sequences recursively 1 so,...
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5.6 Defining Sequences Recursively 1
Discrete Structures
Chapter 5: Sequences, Mathematical Induction, and Recursion
5.6 Defining Sequences Recursively
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
5.6 Defining Sequences Recursively 2
Definition
• Recurrence Relation
A recurrence relation for a sequence a0, a1, a2, … is a formula that relates each term ak, to certain of its predecessors ak-1, ak-2, …, ak-i where i is an integer with k – i 0.
• Initial Conditions
The initial conditions for such a recurrence relation specify the values of a0, a1, a2, …, ai-1, if i is a fixed integer, or a0, a1, a2, …, am where m is an integer with m 0, if i depends on k.
5.6 Defining Sequences Recursively 3
Example – pg. 302 #2
• Find the first four terms of the recursively defined sequence.
1
1
3 , for all integers 2
1k kb b k k
b
5.6 Defining Sequences Recursively 4
Example – pg. 302 #10
0 1 2
1
Let , , ,... be defined by the formula
4 , for all integers 0. Show that
this sequence satisfies the recurrence
relation 4 for all integers 1.
nn
k k
b b b
b n
b b k
5.6 Defining Sequences Recursively 5
Fibonacci Numbers
• 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 Recursively 6
Fibonacci Numbers
• The solution is a recurrence relation
1 2
0 1
(1)
(2) 1, 1k k kF F F
F F
5.6 Defining Sequences Recursively 7
Tower of Hanoi
• Please read this section in your textbook.
5.6 Defining Sequences Recursively 8
Example – pg. 303 #28
• F0, F1, F2, … is the Fibonacci sequence.
2 2 21 1 1Prove that 2 ,
for all integers 1.k k k k kF F F F F
k
5.6 Defining Sequences Recursively 9
Definition
Given numbers a1, a2, …, an, where n is a positive integer,
• the summation from i = 1 to n of the ai is defined as follows:
if n > 1.
• the product from i = 1 to n of the ai is defined by:
if n > 1.
1 1
1 1 1
and n n
i i i i ni i i
a a a a a
1 1
1 1 1
and n n
i i i i ni i i
a a a a a
5.6 Defining Sequences Recursively 10
Example – pg. 304 #42
1 2 1 2
1
Use the recursive definition of product,
together with mathematical induction, to
prove that for all positive integers , if
, , ..., and , , ..., are real
numbers, then
n n
n
i ii
n
a a a a a a
a b 1 1
.n n
i ii i
a b