Discrete Structures

1. Discrete Structures Chapter 5: Sequences, Mathematical Induction, and Recursion 5.1 Sequences A mathematician, like a painter or poet, is a maker of patterns. – G. H. Hardy, 1877 – 1947 A Mathematician’s Apology, 1940 5.1 Sequences

2. Definitions • Sequence A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer. We typically write a sequence as • Term Each individual element ak is called a term and is read “a sub k”. • Subscript The k in ak is called a subscript or index. • Initial Term The m may be any integer and is the subscript of the initial term. • Final Term The n must be greater than or equal to m and is the subscript of the final term. 5.1 Sequences

3. Definitions • An infinite sequence is denoted by • An explicit formula or general formula for a sequence is a rule that shows how the values of ak depend on k. 5.1 Sequences

4. Example – pg. 242 #1 • Write the first four terms of the sequences below. 5.1 Sequences

5. Example – pg. 243 # 11 • Find explicit formulas for sequences of the form a1, a2, a3, … with the initial terms given. 5.1 Sequences

6. Definitions 5.1 Sequences

7. Definition 5.1 Sequences

8. Theorem 5.1.1 5.1 Sequences

9. Examples – pg. 243 • Compute the summations and products. 5.1 Sequences

10. Example • Write using summation or product notation. 5.1 Sequences

11. Definitions 5.1 Sequences

12. Formula 5.1 Sequences

13. Examples – pg. 243 • Compute each of the following. Assume the values of the variables are restricted so that the expressions are defined. 5.1 Sequences