1. Section 11.1 Sequences and Series
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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