1. A Brief Summary for Exam 2
Subject Topics
Mathematical Induction & Recursion (sections )
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

2. 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)

3. 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.

4. 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

5. 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

6. 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