No.25

Permutation: order matters. Example, 12 candidates running for one president, one vice president and one secretary. There are 12*11*10 ways. Because it asks specific orders.
Combination: order does not matter. Example, 12 candidates running for three positions. How many ways? In this case, since order does not matter, use combination. 12C3.

No.24

Mathematical induction:
Step1. Prove n=1 is true
             Left hand side equal to the first number in the sequence
             Right hand side equal to the number when you plug n=1in the other side of the equation
             If left is equal to right, then the statement is true
Step2. Assume n=1is true. Prove n+1is true
             Left hand side is the sequence when you remove n by n+1
             Right hand side is the sequence when you also plug n+1
             If they are equal, the statement is true. Hence,

No.23

This week, we learned an interesting math topic, the Pascal's triangle, which was named after the French mathematician Blaise Pascal. However, Chinese and Indian mathematicians also found this rule decades before him. The Chinese mathematician Yang Hui was a famous guy in Song dynasty. He worked on magic squares, magic circles and binomial theorems. He was well known as the founder of the "Yang Hui's Triangle".(same as Pascal's.)

No.24

Every nonempty set of nonnegative integers has at least element 

• Let C = {n: P(n) is false} (the set of “counterexamples”)
• •Assume C is nonempty in order to derive a contradiction
• •Let m be the smallest element of C
• •Derive a contradiction (perhaps by finding a smaller member of C)
• 
  

No.21

Mathematical induction is a form of mathematical proof. There are two steps of mathematical induction:
1. Prove the statement is true at the starting point.
2. Assume the statement is true for n. Prove the statement is true for n+1.
Example: 1+3+5+7+...+(2n-1)=n^2
Step1 n=1
    LHS=1
    RHS=1^2=1
Since LHS=RHS is true for n=1
Step2 assume true for n. Show true for n+1
1+3+5+...+(2n+1)=(n+1)^2
    LHS=1+3+5+...+(2n-1)+(2n+1)
           = n^2+ 2n+1
    RHS=(n+1)^2
Since LHS=RHS is true for n+1
Hence the statement is true for all n element of natural number.

No.20

In today's lesson, we learned how to find the common difference of arithmetic sequences, which should be a review lesson for most of use. Arithmetic sequence is defined as consecutive terms have a common difference. For example, 2,4,6,8,10....
the common difference for this should be 2. Since the term is always 2 more than the previous one.

No.19

Today we reviewed the lesson of sequences in Algebra 2. A sequence is a function whose domain is natural numbers. Rather than using function notations, they are usually written in  a_n  form.  Here is a youtube clip about intro to sequences.