Arithmetic and Geometric Sequences

1. Arithmetic and Geometric Sequences by Pam Tobe Beth Bos Mary Lou Shelton

2. Suppose you have $40 in a piggy bank that you are saving to spend on a special project. You take on a part-time job that pays $13 per day. Each day you put the cash into the piggy bank. The number of dollars in the bank is a function of the number of days you have worked.

3. But it is a discrete function rather than a continuous function. After 3 ½ days you still have the same $79 as you did after 3 days. A function like this, whose domain is the set of positive integers, is called a sequence.

4. Objectives: • Represent sequences explicitly and recursively Given information about a sequence • Find a term when given its term number • Find the term number of a given term.

5. Let’s investigate the sequence of dollars 53, 66, 79, 92, 105… in the previous problem by: • Sketching the graph of the first few terms of the sequence. • Finding t100n the 100th term of the sequence • Writing an equation for tn the nth term of the sequence, in terms of n.

6. a) The graph shows discrete point. You may connect the points with a dashed line to show the pattern, but don’t make it a solid line because sequences are defined on the set of natural numbers.

7. b. To get the fourth term, you add the common difference of 13 three times to 53. So to get the 100th term, you add the common difference 99 times to 53. t100n = 53 + 99(13) = 1340 c. tn = 53 +13(n-1)

8. The sequence in Example 1 is called an arithmetic sequence. You get each term by adding the same constant to the preceding term. You can also say that the difference of consecutive terms is a constant. The constant is called the common difference.

9. The pattern “add 13 to the previous term to get the next term” in Example 1 is called a recursive pattern for the sequence. You can write an algebraic recursion formula tn = tn-1 + 13 Sequence mode: nMin = 1 beginning value of n u(n) = u(n-1)+13 recursion formula u(uMin) = {53} enter first term Press Graph

10. The pattern tn = 53 + 13(n-1) is called an explicit formula for the sequence. It “explains” how to calculate any desired term without finding the terms before it.

11. Arithmetic Sequence:A sequence in which consecutive terms differ by a fixed amount is an arithmetic sequence, or arithmetic progression.

12. Pair/Share, Try It Determine whether the sequence could be arithmetic. If so, find the common difference. • -6, -3.5, -1, 1.5, 4,…… • 48, 24, 12, 6, 3,….. • In3, In 5, In12, In24

13. If (an ) is an arithmetic sequence with common difference d, then a2 = a1 + d a3 = a2 + d = a1 + 2d a4 = a3 + d = a1 + 3d

14. Pair/Share, Try It The third and eighth terms of an arithmetic sequence are 13 and 3, respectively. Find the first term, the common difference, and an explicit rule for the nth term.

15. Geometric Sequence In an arithmetic sequence, terms are found by adding a constant to the preceding term. A sequence in which terms are found by multiplying the preceding term by a (nonzero) constant is a geometric sequence or geometric progression.

17. Pair/Share, Try It Determine whether the sequence could be geometric. If so, find the common ratio.

19. Application • The population of Bridgetown is growing at the rate of 2.5% per year. The present population is 50,000. Find a sequence that represents Bridgetown’s population each year. Represent the nth term of the sequence both explicitly and recursively. Evaluate seven terms of the sequence.

20. Assume that the population of New York and Texas continued to grow at the annual rate as shown: • In what year will the population of Texas surpass that of New York? • In what year will the population of Texas surpass that of New York by 1 million?