Download presentation
Presentation is loading. Please wait.
1
Section 11.1 Sequences and Series
Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.
2
Objectives Find terms of sequences given the nth term.
Look for a pattern in a sequence and try to determine a general term. Convert between sigma notation and other notation for a series. Construct the terms of a recursively defined sequence.
3
Sequences A sequence is a function, where the domain is a set of consecutive positive integers beginning with 1. An infinite sequence is a function having for its domain the set of positive integers, {1, 2, 3, 4, 5, …}. A finite sequence is a function having for its domain a set of positive integers, {1, 2, 3, 4, 5, …, n}, for some positive integer n.
4
Sequence Formulas In a formula, the function values are known as terms of the sequence. The first term in a sequence is denoted as a1, the fifth term as a5 , and the nth term, or the general term, as an.
5
Example Predict the general term of the sequence 4, 16, 64, 256, … These are the powers of 4, so the general term might be (4)n.
6
Example Find the first 4 terms and the 9th term of the sequence whose general term is given by an = 4(2)n. We have an = 4(2)n, so a1 = 4(2)1 = 8 a2 = 4(2)2 = 16 a3 = 4(2)3 = 32 a4 = 4(2)4 = 64 a9 = 4(2)9 = 2048
7
Alternating Sequence The power (2)n causes the sign of the terms to alternate between positive and negative, depending on whether the n is even or odd. This kind of sequence is called an alternating sequence.
8
Sums and Series
9
Example For the sequence 1, 3, 5, 7, 9, 11, 13, … find each of the following: a) S1 b) S5 c) S7 Solution: a) S1 = 1 b) S5 = (5) (9) = 5 c) S7 = (5) (9) (13) = 7
10
Sigma Notation The Greek letter (sigma) can be used to simplify notation when the general term of a sequence is a formula. For example, the sum of the first three terms of the sequence ,…, ,… can be named as follows, using sigma notation, or summation notation: This is read “the sum as k goes from 1 to 3 of .” The letter k is called the index of summation. The index of summation might be a number other than 1, and a letter other than k can be used.
11
Example Find and evaluate the sum. Solution: = 9 + (27) + 81 = 6
12
Example Write sigma notation for the sum … Solution: … = … This in an infinite series, so we use the infinity symbol to write the sigma notation.
13
Recursive Definitions
A sequence may be defined recursively or by using a recursion formula. Such a definition lists the first term, or the first few terms, and then describes how to determine the remaining terms from the given terms.
14
Example Find the first 5 terms of the sequence defined by
15
Example Find the first 4 terms of the sequence defined by a1 = 3, an+1= 3an 2 for n 1. a1 = 3 a2 = 3a1 2 = 3 3 2 = 7 a3 = 3a2 2 = 3 7 2 = 19 a4 = 3a3 2 = 3 19 2 = 55
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.