Chapter 11 Sequences, Induction, and Probability Copyright © 2014, 2010, 2007 Pearson Education, Inc Sequences and Summation Notation

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Find particular terms of a sequence from the general term. Use recursion formulas. Use factorial notation. Use summation notation. Objectives:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Definition of a Sequence An infinite sequence {a n } is a function whose domain is the set of positive integers. The function values, or terms, of the sequence are represented by a 1, a 2, a 3, a 4,..., a n,... Sequences whose domains consist only of the first n positive integers are called finite sequences.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Writing Terms of a Sequence from the General Term Write the first four terms of the sequence whose nth term, or general term, is given: To find the first four terms, we replace n in the formula with 1, 2, 3, and 4. The first four terms of the sequence are 7, 9, 11, 13.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Writing Terms of a Sequence from a General Term Write the first four terms of the sequence whose nth term, or general term, is given: To find the first four terms of the sequence, we replace n in the formula with 1, 2, 3, and 4.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Writing Terms of a Sequence from a General Term (continued) Write the first four terms of the sequence whose nth term, or general term, is given: The first four terms of the sequence are

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Recursion Formulas A recursion formula defines the nth term of a sequence as a function of the previous term.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Using a Recursion Formula Find the first four terms of the sequence in which a 1 = 3 and for The terms are 3, 11, 27, 59.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Factorial Notation If n is a positive integer, the notation n! (read “n factorial”) is the product of all positive integers from n down through 1. 0! (zero factorial), by definition is 1. 0! = 1

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Finding Terms of a Sequence Involving Factorials Write the first four terms of the sequence whose nth term is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Finding Terms of a Sequence Involving Factorials (continued) Write the first four terms of the sequence whose nth term is The first four terms of the sequence are

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Summation Notation The sum of the first n terms of a sequence is represented by the summation notation where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Using Summation Notation Expand and evaluate the sum:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Using Summation Notation Expand and evaluate the sum:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Using Summation Notation Expand and evaluate the sum: