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 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 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.

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 = -1 + 5n.

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.

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

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.

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:

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).

The Binomial Theorem - produces

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

The Binomial Theorem The expansion of (x+y)n is: Example: (x+y)2 = x2 + 2xy + y2 (x+y)3 = x3 + 3x2y + 3xy2 + y3

The Binomial Theorem Develop (x + y) 8 (x + y)8 = x8 + 8 x7 y + 28 x6 y2 + 56 x5 y3 + 70 x4 y4 + 56 x3 y5 + 28 x2 y6 + 8 x y7 + y8