1 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac Lecture 09: SEQUENCES Section 3.2 Jarek Rossignac CS1050: Understanding and Constructing Proofs Spring 2006

2 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac Lecture Objectives Analyze/evaluate sequences

3 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac What is a sequence? A function that maps an element n in the set {0,1,2…} or {1,2,3…} into the term a n in an ordered set S. Notation {a n } describes the sequence. For example {1/n} is {1, 1/2, 1/3,…}

4 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac What is an arithmetic progression? {a+nd} a = initial term d = common difference Examples: {1, 3, 5, 7…} {(–1) n } = ?

5 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac What is a geometric progression? {ar n } a = initial term d = common ratio Examples: {1, 2, 4, 8…} {(–1) n } = ?

6 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac How to find the formula for a sequence (a)1, –1/2, 1/4, –1/8… a n =? (b)1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …? (c)1, 8, 27… a n =? (d)3, 9, 27, 81… a n =? (e)1, 7, 25, 79, 241… a n =?

7 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac Next term?

8 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac How to compute sums of sequences? ∑ k=0 n (k) = (n+1)n/2 ∑ k=0 n (r k ) = (r n+1 –1)/(r–1) for r≠1 ∑ k=0  (x k ) = 1/(1–x) for |x|<1

9 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac What is a countable set? A and B have the same cardinality if there is a bijection between them. A set is countable is it has the same cardinality as the set of positive integers. –Positive rational numbers are countable –Real numbers are not

10 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac Assigned Homework Page : 9g and 28

11 Georgia Tech, IIC, GVU, 2006 MAGIC Lab Rossignac Assigned Project