This document introduces summation notation and how to evaluate series expressed in sigma notation. It explains that a series represents the sum of terms in a sequence and defines partial sums. Summation notation uses the capital sigma (Σ) symbol to represent the sum of terms defined by a rule. The document provides examples of writing series in summation notation and evaluating them. It also derives formulas for finding the sums of constant, linear, quadratic, and other common series without having to expand them out.

1. UNIT 12 SUM ANDUNIT 12 SUM AND
SIGMA NOTATIONSIGMA NOTATION

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

5. series
partial sum
summation notation
Vocabulary

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 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 Σ (capital sigma) to denote the sum of
a sequence defined by a rule, as shown.

10. Caution!

11. 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.

12. 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.

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

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

15. 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.
The formula for the sum of a constant series is
as shown.

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

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

18. Similar methods will help you find the sum of a
quadratic series.
Notice that 5 is half of the number of terms and
11 represents 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

20. 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

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

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

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

24. 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?

25. 11 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.

26. 22 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.

27. Solve33
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.

28. Look Back44
Use the diagram to continue the pattern.
The 6th loop would be 294 inches.
S6 = 34 + 40 + 46 + 52 + 58 + 64 = 294.

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