1. Series and Summation Notation
12-2
Series and Summation Notation
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

2. Warm Up
Find the first 5 terms of each sequence. 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
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 series is 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.
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 –
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) =
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. Finding the sum of a series with many terms can be tedious
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 , each term has the same value.
The formula for the sum of a constant series is as shown.

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 5 is half of the number of terms and 11 represents the sum of the first and the last term, 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. 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
Caution

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. Method 2 Expand and evaluate. Method 1 Use the summation formula.
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. =
4 items
= nc = 60(4) = 240
= 240

29. Check It Out! Example 3b
Evaluate each series.
Linear series
Method 1 Use the summation formula.
Method 2 Expand and evaluate.
= = 120

30. Check It Out! Example 3c
Evaluate the series.
Quadratic series
Method 1 Use the summation formula.
Method 2 Use a graphing calculator.
n(n + 1)(2n + 1) 6 =
10(10 + 1)(2 · ) 6 =
(110)(21) 6 =
= 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. Understand the Problem1
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. 2
Make a Plan
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. Solve 3
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. Solve 3
Where k is the row number and n is the total number of rows.
Evaluate the series for several n-values.
2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10) =
= 110
2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10) + 2(11) =
= 132

36. Solve 3
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 =
= 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. Understand the Problem1
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. 2
Make a Plan
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. Solve 3
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 = = 294.

43. Lesson Quiz: Part I
Write each series in summation notation.
1. 1 – – ,000 2.
Write each series in summation notation. 55 3. 5.
325 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