Sequences and Series (T) Students will know the form of an Arithmetic sequence.  Arithmetic Sequence: There exists a common difference (d) between each term.  Ex: 2, 6, 10, 14,…d = ?  Ex: 17, 10, 3, -4, -11,…d = ?  Ex: a, a+d, a+2d, a+3d, a+4d,…  General term:

(T) Students will know the form of a geometric sequence.  Geometric Sequence: There exists a common ratio (r) between each term.  Ex: 1, 3, 9, 27, 81 r = ?  Ex: 64, -32, 16, -8, 4 r = ?  Ex: a, ar, ar^2, ar^3, ar^4,…  General term:

(T) Students know the recursive definition.  Recursive definitions: The next value of the sequence is determined using the previous term.  Ex: Explicit definitions were given in the previous section.

 Ex: 23, 20, 17, 14,…  Recursive def:  Explicit def:

(T) Students understand the concept of a series.  Arithmetic and Geometric Series: A series is the sum of the terms of a sequence.  Finite sum of an arithmetic series: Illustrate proof:

(T) Students can find the sum of a series.  Sum of a finite geometric series: Where r can not equal 1. Illustrate proof.

(T) Students understand the concept of limits.  Infinite Sequences: A sequence that continues forever.  Ex: ½, ¼, 1/8, 1/16, …, (1/2)^n,…  Limits: The sequence approaches some number but never reaches it. On a graph it is an asymptote.  Ex:

 Theorem:  (T) Students know the formula for an infinite geom. Series.  Sum of an infinite geometric series:

(T) Students can use mathematical induction for proofs.  Mathematical Induction: Let S be a statement in terms of a positive integer n.  Show that S is true for n=1  Assume that S is true for n=k, where is a positive integer, and then prove that S must be true for n = k + 1. Prove that Prove that n^3 + 2n is a multiple of 3.