5.7 Arithmetic and Geometric Sequences 275 Under optimal laboratory conditions, the number of Escherichia coli (E. coli ) bacteria will double every 17 minutes. If an experiment conducted under optimal conditions begins with 100 bacteria, how many bacteria will be present after 24 hours? Questions such as these are important to the modern study of diseases. In this section, we will study such questions by using lists of numbers known as sequences. Arithmetic and Geometric Sequences SECTION 5.7 LEARNING GOALS Upon completion of this section, you will be able to: 7 Identify and solve problems involving arithmetic sequences. 7 Identify and solve problems involving geometric sequences. Why This Is Important Sequences are used throughout many branches of mathematics to model real-life phenomena, such as population growth, disease control, and the inflation rates of currencies. A sequence is a list of numbers that are related to each other by a rule. The numbers that form the sequence are called its terms. If your salary increases or decreases by a fixed amount over a period of time, the listing of the amounts, over time, would form an arithmetic sequence. When interest in a savings account is compounded at regular intervals, the listing of the amounts in the account over time will be a geometric sequence. Arithmetic Sequences A sequence in which each term after the first term differs from the preceding term by a constant amount is called an arithmetic sequence. The amount by which each pair of successive terms differs is called the common difference, d. The common difference can be determined by subtracting any term from the term that directly follows it. Examples of arithmetic sequences Common differences … 1, 5, 9, 13, 17, d 5 1 4 = − = − − − − … 7, 5, 3, 1, 1, = − − − = − + = d 5 ( 7) 5 7 2 − … 5 2 , 3 2 , 1 2 , 1 2 , d 3 2 5 2 2 2 1 = − = − = − Example 1 The First Five Terms of an Arithmetic Sequence Write the first five terms of an arithmetic sequence with first term 6 and common difference of 4. Solution The first term is 6. The second term is 6 4, + or 10. The third term is 10 4, + or 14. The fourth term is 14 4, + or 18. The fifth term is 18 4, + or 22. Thus, the first five terms of the sequence are 6, 10, 14, 18, 22. 7 Now try Exercise 7 Example 2 An Arithmetic Sequence with a Negative Difference Write the first five terms of the arithmetic sequence with first term 9 and a common difference of 4. − Solution The first five terms of the sequence are − − 9, 5, 1, 3, 7 7 Now try Exercise 9 Kateryna Kon/ Shutterstock
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