1. Sequences and the Binomial Theorem

2. Definitions
Patterns are useful to predict what came before or what might come after a set a numbers that are arranged in a particular order.
This arrangement of numbers is called a sequence.
For example: 3,6,9,12 and 15 are numbers that form a pattern called a sequence.
The numbers that are in the sequence are called terms.

3. Sequences
Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2, } to a set S.
A sequence is an ordered list of numbers: 2,5,7, …
Note: the sets {0, 1, 2, 3, , k} and {1, 2, 3, 4, , k} are called initial segments of N.
Notation: if f is a function from {0, 1, 2, . . .} to S we usually denote f(i) by ai and we write
where k is the upper limit (usually ).

4. Sequences
Examples: Using zero-origin indexing, if f(i) = 1/(i + 1). then the sequence f = {1, 1/2,1/3,1/4, } = {a0, a1, a2, a3, … } Using one-origin indexing the sequence f becomes {1/2, 1/3, . . .} = {a1, a2, a3, . . .}
Some sequences are finite (they have a last term), others are infinite (they do not have a last term).
The first term is generally a1.
The general term, or nth term, is an.

5. Sequences
Definition: An arithmetic progression (sequence) is a sequence of the form
a, a+d, a+2d, a+3d , , where d is the common difference.
General term: an = a1 + (n-1)d.
Example: find the nth term of the arithmetic sequence: 4,9,14, … an = 4 + (n-1)5 = n.

6. Sequences
Definition: A geometric progression is a sequence of the form a, ar, ar 2 , ar3 , ar4 , . . .
General term: an = a1 rn-1.
Example: find the nth term of the arithmetic progression: 4, 8, 16, 32, … an = 4 . 2n-1.

7. The Binomial Theorem Combinations
Selection is without replacement but order does not matter .
It is equivalent to selecting subsets of size r from a set of size n.
The number of combinations of n things taken r at a time

8. The Binomial Theorem Other names for C(n,r):
n choose r
The binomial coefficient
How many subsets of size r can be constructed from a set of n objects?
The answer is clearly C(n,r) since once we select the objects (without replacement) the order doesn't matter.

9. The Binomial Theorem
Corollary: Proof: If we count the number of subsets of a set of size n, we get the cardinality of the power set.
Properties:

10. The Binomial Theorem Pascal's Identity: Proof:
We construct subsets of size k from a set with n + 1 elements given the subsets of size k and k-1 from a set with n elements.
The total will include
all of the subsets from the set of size n which do not contain the new element C(n, k),
the subsets of size k - 1 with the new element added C(n, k-1).

11. The Binomial Theorem - produces

12. The Binomial Theorem
Give the row of Pascal’s triangle immediately following:

13. The Binomial Theorem The expansion of (x+y)n is:
Example: (x+y)2 = x2 + 2xy + y (x+y)3 = x3 + 3x2y + 3xy2 + y3

14. The Binomial Theorem
Develop (x + y) 8 (x + y)8 = x8 + 8 x7 y + 28 x6 y x5 y x4 y x3 y x2 y6 + 8 x y y8