Series and Summation Notation

1. Series and Summation Notation Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

2. Warm Up Find the first 5 terms of each sequence. 1.4. 2. 5. 3.

3. Warm Up Continued Write a possible explicit rule for the nth term of each sequence. 6. 1, 2, 4, 8, 16, … 7. 4, 7, 10, 13, 16, …

4. Objective Evaluate the sum of a series expressed in sigma notation.

5. Vocabulary series partial sum summation notation

6. In Lesson 12-1, you learned how to find the nth term of a sequence. Often we are also interested in the sum of a certain number of terms of a sequence. A seriesis the indicated sum of the terms of a sequence. Some examples are shown in the table.

7. Because many sequences are infinite and do not have defined sums, we often find partial sums. A partial sum, indicated by Sn, is the sum of a specified number of terms of a sequence.

9. A series can also be represented by using summation notation, which uses the Greek letter  (capital sigma) to denote the sum of a sequence defined by a rule, as shown.

10. Example 1A: Using Summation Notation Write the series in summation notation. 4 + 8 + 12 + 16 + 20 Find a rule for the kth term of the sequence. ak = 4k Explicit formula Write the notation for the first 5 terms. Summation notation

11. Example 1B: Using Summation Notation Write the series in summation notation. Find a rule for the kth term of the sequence. Explicit formula. Write the notation for the first 6 terms. Summation notation.

12. Caution!

13. Check It Out! Example 1a Write each series in summation notation. Find a rule for the kth term of the sequence. Explicit formula. Write the notation for the first 5 terms. Summation notation.

14. Check It Out! Example 1b Write the series in summation notation. Find a rule for the kth term of the sequence. Explicit formula. Write the notation for the first 6 terms. Summation notation.

15. Example 2A: Evaluating a Series Expand the series and evaluate. Expand the series by replacing k. Evaluate powers. Simplify.

16. Example 2B: Evaluating a Series Expand the series and evaluate. = (12 – 10) + (22 – 10) + (32 – 10) + (42 – 10) + (52 – 10) + (62 – 10) Expand. = –9 – 6 – 1 + 6 + 15 + 26 Simplify. = 31

17. Check It Out! Example 2a Expand each series and evaluate. Expand the series by replacing k. = (2(1) – 1) + (2(2) – 1) + (2(3) – 1) + (2(4) – 1) = 1 + 3 + 5 + 7 Simplify. = 16

18. Check It Out! Example 2b Expand each series and evaluate. Expand the series by replacing k. = –5(2)(1– 1) – 5(2)(2– 1) – 5(2)(3– 1) – 5(2)(4– 1) – 5(2)(5– 1) = – 5 – 10 – 20 – 40 – 80 Simplify. = –155

19. The formula for the sum of a constant series is as shown. Finding the sum of a series with many terms can be tedious. You can derive formulas for the sums of some common series. In a constant series, such as 3 + 3 + 3 + 3 + 3, each term has the same value.

20. The formula for the sum of a constant series is as shown.

21. A linear series is a counting series, such as the sum of the first 10 natural numbers. Examine when the terms are rearranged.

22. Notice that 5is half of the number of terms and 11represents the sum of the first and the last term, 1 + 10. This suggests that the sum of a linear series is , which can be written as Similar methods will help you find the sum of a quadratic series.

24. Caution When counting the number of terms, you must include both the first and the last. For example, has six terms, not five. k = 5, 6, 7, 8, 9, 10

25. Example 3A: Using Summation Formulas Evaluate the series. Constant series Method 1 Use the summation formula. Method 2 Expand and evaluate. There are 7 terms.

26. Example 3B: Using Summation Formulas Evaluate the series. Linear series Method 1 Use the summation formula. Method 2 Expand and evaluate.

27. Example 3C: Using Summation Formulas Evaluate the series. Quadratic series Method 1 Use the summation formula. Method 2 Use a graphing calculator.

28. 4 items = nc = 60(4)= 240 Check It Out! Example 3a Evaluate the series. Constant series Method 2 Expand and evaluate. Method 1 Use the summation formula. There are 60 terms. = 60 + 60 + 60 + 60 = 240

29. Check It Out! Example 3b Evaluate each series. Linear series Method 1 Use the summation formula. Method 2 Expand and evaluate. = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 120

30. n(n + 1)(2n + 1) 6 = (110)(21) 6 = 10(10 + 1)(2 ·10 + 1) 6 = Check It Out! Example 3c Evaluate the series. Quadratic series Method 1 Use the summation formula. Method 2 Use a graphing calculator. = 385

31. Example 4: Problem-Solving Application Sam is laying out patio stones in a triangular pattern. The first row has 2 stones and each row has 2 additional stones, as shown below. How many complete rows can he make with a box of 144 stones?

32. 1 Understand the Problem The answer will be the number of complete rows. List the important information: • The first row has 2 stones. • Each row has 2 additional stones • He has 144 stones. • The patio should have as many complete rows as possible.

33. Make a Plan 2 Make a diagram of the patio to better understand the problem. Find a pattern for the number of stones in each row. Write and evaluate the series.

34. 3 Solve Use the given diagram to represent the problem. The number of stones increases by 2 in each row. Write a series to represent the total number of stones in n rows.

35. 3 Solve Where k is the row number and n is the total number of rows. 2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10) = = 2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10) + 2(11) Evaluate the series for several n-values. = 110 = 132

36. 3 Solve 2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10) + 2(11) + 2(12) = 156 Because Sam has only 144 stones, the patio can have at most 11 complete rows.

37. Look Back 4 Use the diagram to continue the pattern. The 11th row would have 22 stones. S11 = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 = 132 The next row would have 24 stones, so the total would be more than 144.

38. Check It Out! Example 4 A flexible garden hose is coiled for storage. Each subsequent loop is 6 inches longer than the preceding loop, and the innermost loop is 34 inches long. If there are 6 loops, how long is the hose?

39. 1 Understand the Problem The answer will be the total length of the hose. List the important information: • The first loop is 34 inches long. • Each subsequent loop is 6 inches longer than the previous one. • There are 6 loops.

40. Make a Plan 2 Make a diagram of the hose to better understand the problem. Find a pattern for the length of each loop. Write and evaluate the series.

41. 3 Solve Use the given diagram to represent the problem. The first loop is 34 in. Each subsequent loop increases by 6 in. (34 + 6(1 – 1)) + (34 + 6(2 – 1)) + (34 + 6(3 – 1)) + (34 + 6(4 – 1)) + (34 + 6(5 – 1)) + (34 + 6(6 – 1)) = 294 in.

42. Look Back 4 Use the diagram to continue the pattern. The 6th loop would be 294 inches. S6 = 34 + 40 + 46 + 52 + 58 + 64 = 294.

43. Lesson Quiz: Part I Write each series in summation notation. 1. 1 –10 + 100 –1000 + 10,000 2. Write each series in summation notation. 55 325 3. 5. 64 4. 6. 285

44. Lesson Quiz: Part II 7. Ann is making a display of hand-held computer games. There will be 1 game on top. Each row will have 8 additional games. She wants the display to have as many rows as possible with 100 games. How many rows will Ann’s display have? 5