1. 7.2 - 1 10 TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA

2. 7.2 - 2 7.2 Arithmetic Sequences and Series Arithmetic Sequences Arithmetic Series

3. 7.2 - 3 Arithmetic Sequences A sequence in which each term after the first is obtained by adding a fixed number to the previous term is an arithmetic sequence (or arithmetic progression). The fixed number that is added is the common difference. The sequence is an arithmetic sequence since each term after the first is obtained by adding 4 to the previous term.

4. 7.2 - 4 Arithmetic Sequences That is, and so on. The common difference is 4.

5. 7.2 - 5 Arithmetic Sequences If the common difference of an arithmetic sequence is d, then by the definition of an arithmetic sequence, for every positive integer n in the domain of the sequence. Common difference d

6. 7.2 - 6 Example 1 FINDING THE COMMON DIFFERENCE Find the common difference, d, for the arithmetic sequence Solution We find d by choosing any two adjacent terms and subtracting the first from the second. Choosing – 7 and – 5 gives Choosing – 9 and – 7 would give the same result. Be careful when subtracting a negative number.

7. 7.2 - 7 Example 2 FINDING TERMS GIVEN a 1 AND d Find the first five terms for each arithmetic sequence. Solution a. The first term is 7, and the common difference is – 3. Start with a 1 = 7. Add d = – 3. Add – 3.

8. 7.2 - 8 Example 2 FINDING TERMS GIVEN a 1 AND d Find the first five terms for each arithmetic sequence. Solution b. a 1 = – 12, d = 5 Start with a 1. Add d = 5.

9. 7.2 - 9 Look At It This Way If a 1 is the first term of an arithmetic sequence and d is the common difference, then the terms of the sequence are given by and, by this pattern,

10. 7.2 - 10 nth Term of an Arithmetic Sequence In an arithmetic sequence with first term a 1 and common difference d, the nth term, is given by

11. 7.2 - 11 Example 3 FINDING TERMS OF AN ARITHMETIC SEQUENCE Find a 13 and a n for the arithmetic sequence – 3, 1, 5, 9, … Solution Here a 1 = – 3 and d = 1 – (– 3) = 4. To find a 13 substitute 13 for n in the formula for the nth term. n = 13 Work inside parentheses first. Let a 1 = – 3, d = 4. Simplify.

12. 7.2 - 12 Example 3 FINDING TERMS OF AN ARITHMETIC SEQUENCE Find a 13 and a n for the arithmetic sequence – 3, 1, 5, 9, … Solution Find a n by substituting values for a 1 and d in the formula for a n. Let a 1 = – 3, d = 4. Simplify. Distributive property

13. 7.2 - 13 Example 4 FINDING TERMS OF AN ARTHEMTIC SEQUENCE Find a 18 and a n for the arithmetic sequence having a 2 = 9 and a 3 = 15. Solution Find d first; d = a 3 – a 2 = 15 – 9 = 6. Since Let a 2 = 9, d = 6.

14. 7.2 - 14 Example 4 FINDING TERMS OF ARTHEMTIC SEQUENCE Find a 18 and a n for the arithmetic sequence having a 2 = 9 and a 3 = 15. Solution d = a 3 – a 2 = 15 – 9 = 6. Then, Formula for a n ; a 1 = 3, n = 18, d = 6.

15. 7.2 - 15 Example 4 FINDING TERMS OF ARTHEMTIC SEQUENCE Find a 18 and a n for the arithmetic sequence having a 2 = 9 and a 3 = 15. Solution d = a 3 – a 2 = 15 – 9 = 6. and Distributive property.

16. 7.2 - 16 Example 5 FINDING THE FIRST TERM OF AN ARITHMETIC SEQUENCE Suppose that an arithmetic sequence has a 8 = – 16 and a 16 = – 40. Find a 1 Solution Since a 16 = a 8 + 8d, it follows that And so d = – 3. To find a 1, use the equation a 8 = a 1 + 7d. Let a 8 = – 16. Let d = – 3.

17. 7.2 - 17 FINDING THE FIRST TERM OF AN ARITHMETIC SEQUENCE Formula for the nth term. Distributive property. The graph of any sequence is a scatter diagram. To determine the characteristics of the graph of an arithmetic sequence, start by rewriting the formula for the nth term. Commutative & associative properties. Let c = a 1 – d.

18. 7.2 - 18 FINDING THE FIRST TERM OF AN ARITHMETIC SEQUENCE Slopey-intercept The points in the graph of an arithmetic sequence are determined by  (n) = dn + c, where n is a natural number. Thus, the points in the graph of  must lie on the line

19. 7.2 - 19 FINDING THE FIRST TERM OF AN ARITHMETIC SEQUENCE For example, the sequence shown here is an arithmetic sequence because the points that comprise its graph are collinear (lie on a line). The slope determined by these points is 2, so the common difference d equals 2.

20. 7.2 - 20 FINDING THE FIRST TERM OF AN ARITHMETIC SEQUENCE On the other hand, the sequence b n shown here is not an arithmetic sequence because the points are not collinear.

21. 7.2 - 21 Example 6 FINDING THE nth TERM FROM A GRAPH Find a formula for the nth term of the sequence a n shown here. What are the domain and range of this sequence? Solution The points in this graph lie on a line, so the sequence is arithmetic. The equation of the dashed line shown here is y = –.5x + 4, so the nth term of this sequence is determined by

22. 7.2 - 22 Example 6 FINDING THE nth TERM FROM A GRAPH Find a formula for the nth term of the sequence a n shown here. What are the domain and range of this sequence? Solution The sequence is comprised of the points

23. 7.2 - 23 Example 6 FINDING THE nth TERM FROM A GRAPH Find a formula for the nth term of the sequence a n shown here. What are the domain and range of this sequence? Solution Thus, the domain of the sequence is given by {1, 2, 3, 4, 5, 6}, and the range is given by {3.5, 3, 2.5, 2, 1.5, 1}.

24. 7.2 - 24 Arithmetic Series The sum of the terms of an arithmetic sequence is an arithmetic series. To illustrate, suppose that a person borrows $3000 and agrees to pay $100 per month plus interest of 1% per month on the unpaid balance until the loan is paid off. The first month, $100 is paid to reduce the loan, plus interest of (.01)3000 = 30 dollars. The second month, another $100 is paid toward the loan, and (.01)2900 dollars is paid for interest

25. 7.2 - 25 Arithmetic Series Since the loan is reduced by $100 each month, interest payments decrease by (.01)100 = 1 dollar each month, forming the arithmetic sequence 30, 29, 28, …, 3, 2, 1.

26. 7.2 - 26 Arithmetic Series The total amount of interest paid is given by the sum of the terms of this sequence. Now we develop a formula to find this sum without adding all 30 numbers directly. Since the sequence is arithmetic, we can write the sum of the first n terms as

27. 7.2 - 27 Arithmetic Series We used the formula for the general term in the last expression. Now we write the same sum in reverse order, beginning with a n and subtracting d.

28. 7.2 - 28 Arithmetic Series Adding respective sides of these two equations term by term, we obtain or since there are n terms of a 1 + a n on the right. Now solve for S n to get

29. 7.2 - 29 Arithmetic Series Using the formula a n = a 1 + (n – 1)d, we can also write this result for S n as or which is an alternative formula for the sum of the first n terms of an arithmetic sequence.

30. 7.2 - 30 Sum of the First n Terms of an Arithmetic Sequence If an arithmetic sequence has first term and common difference d, then the sum of the first n terms is given by or

31. 7.2 - 31 The first formula is used when the first and last terms are known; otherwise the second formula is used. For example, in the sequence of interest payments discussed earlier, n = 30, a 1 = 30, and a n = 1. Choosing the first formula, gives so a total of $465 interest will be paid over the 30 months.

32. 7.2 - 32 Example 7 USING THE SUM FORMULAS Evaluate S 12 for the arithmetic sequence – 9, – 5, – 1, 3, 7, …. a. Solution We want the sum of the first 12 terms. Using a 1 = – 9, n = 12, and d = 4 in the second formula, gives

33. 7.2 - 33 Example 7 USING THE SUM FORMULAS b. Solution Use a formula for S n to evaluate the sum of the first 60 positive integers. The first 60 positive integers form the arithmetic sequence 1, 2, 3, 4, …, 60. Thus, n = 60, a 1 = 1, and a 60 = 60, so we use the first formula in the preceding box to find the sum.

34. 7.2 - 34 Example 8 USING THE SUM FORMULA The sum of the first 17 terms of an arithmetic sequence is 187. If a 17 = – 13, find a 1 and d. Solution Use the first formula for S n, with n = 17. Let S 17 = 187, a 17 = – 13. Multiply by Add 13; rewrite.

35. 7.2 - 35 Example 8 USING THE SUM FORMULA The sum of the first 17 terms of an arithmetic sequence is 187. If a 17 = – 13, find a 1 and d. Solution Let a 17 = – 13, a 1 = 35. Subtract 35. Divide by 16; rewrite. Since a 17 = a 1 + (17 – 1)d,

36. 7.2 - 36 Any sum of the form where d and c are real numbers, represents the sum of the terms of an arithmetic sequence having first term a 1 = d(1) + c = d + c and common difference d. These sums can be evaluated using the formulas in this section.

37. 7.2 - 37 Example 9 USING SUMMATION NOTATION Evaluate each sum. a. Solution This sum contains the first 10 terms of the arithmetic sequence having First term. and Last term. Thus,

38. 7.2 - 38 Example 9 USING SUMMATION NOTATION Evaluate each sum. b. Solution The first few terms are Thus, a 1 = – 5 and d = – 3. If the sequence started with k = 1, there would be nine terms. Since it starts at 3, two of those terms are missing, so there are seven terms and n = 7. Use the second formula for S n.