Recursive Definitions

1
Recursive Definitions

• Section 4.3

2
Mathematical Induction (Recap)

• A proof by mathematical induction that P(n) is
  true for every positive integer n consists of two
  steps
• Basis step The proposition P(1) is shown to be
  true.
• Inductive step The implication P(k)?P(k1) is
  shown to be true for every positive integer k.

3
Strong Induction(Recap)

• To prove that P(n) is true for every positive
  integer n consists of two steps
• Basis step The proposition P(1) is shown to be
  true.
• Inductive step The implication P(1)?P(2)?
  ?P(k)?P(k1) is shown to be true for every
  positive integer k.

4
Mathematical Induction (Recap)

• Use mathematical induction to prove that n3n is
  divisible by 3 whenever n is a positive integer.
• Show that if n is an integer greater than 1, then
  n can be written as the product of primes.

5
Recursion

• We can define a sequence, series, or function in
  terms of itself. This process is called
  recursion.
• To define a function with the set of nonnegative
  integers as its domain
• Specify the value of the function at zero
• Give a rule for finding its value at an integer
  from its values at smaller integers

6
Example

• The sequence of powers of 2 can be written
  explicitly as
• an 2n
• However, this sequence can also be defined using
  recursions as
• a0 1, an1 2an

7
Example

• Suppose that f is defined recursively by
• f(0)3
• f(n1) 2f(n)3
• Find f(1), f(2), f(3), and f(4)

8
Example

• Give a recursive definition of an where a is a
  nonzero real number and n is a nonnegative
  integer.
• Give a recursive definition of the factorial
  function F(n) n!.
• Give a recursive definition for the Fibonacci
  numbers 1 2 3 5 8 13 21 ...