1. BaSIC Math Reviews

2. Math Reviews Mathematical Induction Logarithm Mod Series Derivative
Limit

3. 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

4. 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.

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

6. 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.

7. 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:

8. 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

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

10. 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,…

11. 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) …

12. Limit
Definition:
Properties
L’Hopital’s Rule