Sequences and the Binomial Theorem Definitions Patterns are

Sequences and the Binomial Theorem

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.

Sequences n n 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 ).

Sequences n n Examples: Using zero-origin indexing, if f(i) = 1/(i + 1). then the sequence f = {1, 1/2, 1/3, 1/4, . . . } = {a 0, a 1, a 2, a 3, … } Using one-origin indexing the sequence f becomes {1/2, 1/3, . . . } = {a 1, a 2, a 3, . . . } Some sequences are finite (they have a last term), others are infinite (they do not have a last term). The first term is generally a 1. The general term, or nth term, is an.

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

Sequences n n n Definition: A geometric progression is a sequence of the form a, ar 2 , ar 3 , ar 4 , . . . General term: an = a 1 rn-1. Example: find the nth term of the arithmetic progression: 4, 8, 16, 32, … an = 4. 2 n-1.

The Binomial Theorem n Combinations n n n 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

The Binomial Theorem n Other names for C(n, r): n n 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.

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

The Binomial Theorem n Pascal's Identity: Proof: n n 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 n n 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).

The Binomial Theorem - produces

The Binomial Theorem n Give the row of Pascal’s triangle immediately following: 1 7 21 35 35 21 7 1 1 1 1 1 2 1 3 3 4 6 5 10 6 15 7 21 8 28 1 4 1 10 5 1 20 15 6 1 35 35 21 7 1 56 70 56 28 8 1

The Binomial Theorem The expansion of (x+y)n is: n Example: (x+y)2 = x 2 + 2 xy + y 2 (x+y)3 = x 3 + 3 x 2 y + 3 xy 2 + y 3

The Binomial Theorem n Develop (x + y) 8 (x + y)8 = x 8 + 8 x 7 y + 28 x 6 y 2 + 56 x 5 y 3 + 70 x 4 y 4 + 56 x 3 y 5 + 28 x 2 y 6 + 8 x y 7 + y 8