Copyright © 2011 Pearson Education, Inc. Sequences Section 8.1 Sequences, Series, and Probability

8.1 Copyright © 2011 Pearson Education, Inc. Slide 8-3 Definition: Sequence A finite sequence is a function whose domain is {1, 2, 3, …, n}, the positive integers less than or equal to a fixed positive integer n. An infinite sequence is a function whose domain is the set of all positive integers. Definition

8.1 Copyright © 2011 Pearson Education, Inc. Slide 8-4 The function f(n) = n 2 with domain {1, 2, 3, 4, 5} is a finite sequence. For the independent variable of a sequence, we usually use n (for natural number) rather than x and assume that only natural numbers can be used in place of n. For the dependent variable f(n), we generally write a n (read “a sub n”). So this finite sequence is also defined by a n = n 2 for 1 ≤ n ≤ 5. Definition

8.1 Copyright © 2011 Pearson Education, Inc. Slide 8-5 The terms of the sequence are the values of the dependent variable a n. We call a n the nth term or the general term of the sequence. The equation a n = n 2 provides a formula for finding the n th term. Definition

8.1 Copyright © 2011 Pearson Education, Inc. Slide 8-6 Definition: Factorial Notation For any positive integer n, the notation n! (read “n factorial”) is defined by The symbol 0! is defined to be 1, 0! = 1. In general, n! is the product of the positive integers from 1 through n. The value given to 0! is 1. Factorial Notation

8.1 Copyright © 2011 Pearson Education, Inc. Slide 8-7 A recursion formula gives the nth term as a function of the previous term. If the first term is known, then a recursion formula determines the remaining terms of the sequence. Recursion Formulas

8.1 Copyright © 2011 Pearson Education, Inc. Slide 8-8 An arithmetic sequence can be defined as a sequence in which there is a common difference d between consecutive terms, or it can be defined by giving a general formula that will produce such a sequence. Arithmetic Sequences

8.1 Copyright © 2011 Pearson Education, Inc. Slide 8-9 Definition: Arithmetic Sequence A sequence that has an nth term of the form a n = a 1 + (n – 1)d, where a 1 and d are any real numbers, is called an arithmetic sequence. Note that a n = a 1 + (n – 1)d can be written as a n = dn + (a 1 – d). So the terms of an arithmetic sequence can also be described as a multiple of the term number plus a constant, (a 1 – d). Arithmetic Sequences