Recursive Sequences - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Discrete Mathematics

Lecture 8

Recursive Sequences

• A recurrence relation for a sequence a 0 , a 1 , a 2 , … is a formula that relates each term ak to certain collection of its predecessors. Each recurrence sequence needs initial conditions that make it well-defined
• Famous recurrences: algebraic and geometric sequences, factorial, Fibonacci numbers
• Tower of Hanoi problem
• Compound interest

Solving Recurrences

• Iteration method
• Telescoping
• Range transformation
• Domain transformation
• Recurrences involving sum

Exercises

• Find an explicit formula for:

x (^) k = 3x (^) k-1 + k with x 1 = 1 wk = wk-2 + k with w 1 = 1, w 2 = 2 u (^) k = u (^) k-2 * u (^) k-1 with u 0 = u 1 = 2

Classes of Functions

• Constants
• Polynoms: linear, quadratic
• Exponents
• Logarithms
• Functions in between
• Relationship between different classes

O-notation

• Function f(n) is of order g(n), written f = O(g),

when there exists number M such that there exists number n 0 so that for all n > n 0 we have f(n) <= M

• g(n)

• If f is O(g), then g is Ω(f), or in other words, when

for all numbers M and for all numbers no, there exists n > n 0 such that f(n) > M * g(n)

• If f is O(g) and g is O(f), then we say that f is Θ(g)

or that f and g are of the same order