A Brief Summary for Exam 2 Subject Topics Mathematical Induction & Recursion (sections 3.1 - 3.5) Sequence and summation Definitions (lower/upper limits, double summation) Useful sequences and their summations (arithmetic, geometric, Fibonacci) Induction Definition and relation to natural number Three parts of the proof basis step, inductive hypothesis, inductive step Strong induction

Recursion Basic idea of recursion Recursive definition of Sequences, functions, sets Two parts: base case and recursion Relations to induction Recursive algorithms Pros and cons (wrt iterative algorithms)

Counting (sections 4.1 – 4.5) Useful rules: Sum rule: disjoint, done at different time |A1  A2| = |A1| + |A2| Product rule: disjoint, done at same time |A1  A2| = |A1| * |A2| Inclusion – exclusion rule: overlapping, done at different time |A1  A2| = |A1| + |A2| - |A1  A2| Pigeonhole Principle Idea and rationale at least one box containing at least N/k of the objects.

Permutations and combinations Definitions of permutations, r-permutations, r-combinations Relationship between permutation and combinations Formulae for (P(n,n), P(n, r), and C(n, r) Pascal triangle and Binomial Coefficients

Recurrence Relations (sections 6.1 and 6.2) Definition of recurrence relation and its solution Relationship with recursive definition Ideas of modeling with recurrence relations Ideas of solving linear homogeneous recurrence relation

Types of Questions Conceptual Problem solving Proofs Definitions of terms True/false Multiple choice Problem solving Work with small concrete example problems Proofs Simple theorems or propositions Especially proof by mathematical induction