1 Review for Quiz 1

2 Summations

3 Binomial expansion

Bounding sums Upper bound: Any sum is at most the number of terms times the maximum term – Example: n 2 is at most n*n 2 = n 3 Lower bound: If the terms are non-negative, any sum is at least half the number of terms times the median term – Example: n 2 is at least (n/2)*(n/2) 2 = n 3 /8 4

5 Proving (or disproving) p→q Counterexample (used to prove that p→q is false showing one particular choice of p that makes q false) Direct proof (p→p 1 →…→p n →q ) Contrapositive (prove that ~q→~p) Contradiction (assume p and ~q true, find a contradiction) Induction (prove base case + induction)

6 Induction proof using induction