1. Objective
Evaluate the sum of a series expressed in sigma notation.
Vocabulary
series
partial sum
summation notation

2. In Lesson 9-1, you learned how to find the nth term of a sequence
In Lesson 9-1, 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.

3. 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.

5. 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.

6. 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

7. Caution!

8. 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.

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

10. 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

11. 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.

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

13. 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.

15. 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

16. 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.

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

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

19. 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?

20. 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.

21. 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.

22. 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.

23. 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

24. 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.

25. 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.