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

Download Recursive Sequences - Discrete Mathematics - Lecture Slides and more Slides Discrete Mathematics in PDF only on Docsity! Discrete Mathematics Lecture 8 Docsity.com Recursive Sequences • A recurrence relation for a sequence a0, a1, a2, … 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 Docsity.com Exercises • Find an explicit formula for: xk = 3xk-1 + k with x1 = 1 wk = wk-2 + k with w1 = 1, w2 = 2 uk = uk-2 * uk-1with u0 = u1 = 2 Docsity.com Second-Order Homogenous Recurrences • Second-order homogeneous relation with constant coefficients is a relation of the form: ak = A * ak-1 + B * ak-2, where A and B are constants • Characteristics equation • Distinct roots case: Fibonacci numbers • Single root case: gambler’s ruin Docsity.com Classes of Functions • Constants • Polynoms: linear, quadratic • Exponents • Logarithms • Functions in between • Relationship between different classes Docsity.com