BaSIC Math Reviews

Math Reviews Mathematical Induction Logarithm Mod Series Derivative Limit

Mathematical Induction What is Mathematical induction? A method of proving statements defined for the class of positive integers (or some subset of this class) Proof of a predicate Let P(n) be a predicate defined for integers n, and let a be a fixed integer. Motivation

Mathematical Induction Principle of mathematical induction: 1.Basis Step. Show that P(a) is true. 2.Inductive Step. First, the inductive hypothesis is made - we assume P(k) is true, k > a. Then, using this assumption, we show that P(k+1) must also be true. More formally, we prove the implication, P(k) → P(k+1) for all k > a. If we prove both parts of the definition, we have shown that P(n) is true for all n > a.

Mathematical Induction Example: Prove: 1+4+7…+(3n-2)=(3n2-n)/2

Generalized Mathematical Induction Let P(n) be a predicate defined for integers n, and let a and b be fixed integers with a < b. A proof by generalized mathematical induction has two steps, just like regular induction: Basis Step. Show that P(a), P(a+1), P(a+2), ... , and P(b) are all true. Inductive Step. Show that for any integer k, k > b, if we assume P(i) is true for all integers i with a < i < k (this is the inductive hypothesis), P(k) must also be true. In logical notation the inductive step is: P(a) & P(a+1) & P(a+2) & ... & P(k-1) -> P(k). If a = b, this will be changed to regular induction.

Generalized Mathematical Induction Example: Consider the Fibonacci number: 0,1,1,2,3,5,8,13,21,... Fn=Fn-1+Fn-2 with seed values F0=0, F1=1 Prove:

Logarithm logax = y ↔ ay = x Examples: Properties Log x to base a is y means a raised to the y-th power is x. Examples: log28 = 3, log39 = 2, log1010 = 1, log51 = 0 Properties

Mod m mod n = m – lowerbound(m/n) 9 mod 2 = 1 8 mod 3 = 2

Series Arithmetic series Geometric series Harmonic series 1,3,5,7,… 2,5,8,11,… Geometric series 2,6,18,54,… 1,4,16,64,… Harmonic series 1,1+1/2,1+1/2+1/3,1+1/2+1/3+1/4,…

Derivative The rate of change… Consider a uniform acceleration: a – acceleration. t – time. D – displacement. D = ½ at2 V – velocity. V = at Properties ( c * f(x) )’ = c * f’(x) …

Limit Definition: Properties L’Hopital’s Rule