Sequences A sequence is a function, where the domain is a set of consecutive positive integers beginning with 1.

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1 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 8: Sequences, Series, and Combinatorics Section 8.1 Sequences and Series Sequences A sequence is a function, where the domain is a set of consecutive positive integers beginning with 1. Sequences can be either finite or infinite. For example, the sequence of multiples of 2 is infinite, whereas the sequence of days in June is finite. Sequence Terminology and Notation The function values are known as terms of the sequence. As with other functions, the domain element gives the and the range element gives the. The first term in a sequence is denoted as a 1, the fifth term in a sequence is denoted as a 5, etc. The nth term, or the general term, in a sequence is denoted as. Example 1 Given the sequence a n = (-1) n 1 (3n 5), find the first four terms and a 10 and a 15. Example 2 Given, find a 80. Example 3 Predict the general term, or nth term, a n, of the sequence. 3, 9, 27, 81, 243, Series The sum of the terms of a sequence is called a series. An infinite series is an expression of the form S = a 1 + a 2 + a a n + A finite series (also called a partial sum) is an expression of the form S n = a 1 + a 2 + a a n. Example 4 Find the indicated partial sums for the sequence: 1, -3, 5, -7, 9, -11, a) S 2 b) S 5

2 Sigma Notation A series can be written using sigma notation (or summation notation) using the Greek letter sigma, Σ. 2 A finite series can be written as An infinite series can be written as S n = a 1 + a 2 + a a n = S = a 1 + a 2 + a a n + = The letter i is called the index of summation and maybe replaced by any other letter. The index of summation does not have to start with the number 1. Example 5 Find and evaluate the sum: Example 6 Write sigma notation: Example 7 Write sigma notation:

3 Section 8.2 Arithmetic Sequences and Series 3 Arithmetic Sequences A sequence in which each term after the first is found by adding the same number to the preceding term is an arithmetic sequence. The fixed number that is added is called the common difference. This difference is denoted by d and is found by taking a term and the term directly before it. Example 1 Example 2 Find the first term and the common difference: 0, -6, -12, -18, -24, Find the first term and the common difference: 1/3, 2/3, 1, 4/3, 5/3, nth Term of an Arithmetic Sequence a n = a 1 + (n 1)d To find d when two consecutive terms are not given, subtract the terms and divide by the distance between them ( formula). Example 3 Find the 11 th term of the arithmetic sequence: 2, 6, 10, 14, Example 4 Given the sequence 0.07, 0.12, 0.17,, what term is the number 1.67? Example 5 Find a 1 when d = 1 and a 7 = 2. Example 6 In an arithmetic sequence, a 3 = -5 and a 9 = 37. Find a 1 and d. Write the first five terms of the sequence. Sum of the First n Terms of an Arithmetic Sequence Example 7 Find the sum of the first 14 terms of the arithmetic sequence 16, 12, 8, 4, Example 8 Find the sum: Example 9 Find the sum:

4 Example 10 An orchestra consists of 8 rows of musicians. The first row has 5 musicians, the second row has 7 musicians, and the third row has 9 musicians. 4 a) How many musicians are in the last row? b) What is the total number of musicians in the orchestra?

5 Section 8.3 Geometric Sequences and Series 5 Geometric Sequences A sequence in which each term after the first is found by multiplying the preceding term by the same number is a geometric sequence. The fixed number that is multiplied by each term to produce the next term is called the common ratio. This ratio is denoted by r and is found by taking a term and it by the term directly before it. Example 1 Find the common ratio of the following geometric sequence: -2/3, 4/9, -8/27, 16/81, nth Term of a Geometric Sequence a n = a 1 r n-1 Example 2 Example 3 Find the 11 th term of the geometric sequence: 1, -3, 9, -27, Find the nth, or general, term of the geometric sequence: 25, 5, 1, Sum of the First n Terms of a Geometric Sequence Example 4 Find the sum: Example 5 Gilberto opened a savings account for his daughter and deposited $1500 on the day she was born. Each year on her birthday, he deposited another $1500. If the account pays 7% interest, compounded annually, how much is in the account at the end of the day on her 12 th birthday?

6 Infinite Geometric Series If the common ratio, r, of a geometric series is a fraction between -1 and 1 ( ), then each successive term will be smaller than the one before. Therefore, the sum of an infinite geometric series approaches a and is said to converge. This limit is given by the formula 6 If the common ratio, r, of a geometric series has absolute value of 1 or greater, then the sum does not approach a limit and the sum will not exist. Example 6 Find the sum, if it exists: Example 7 Find the sum, if it exists: