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

Holt McDougal Algebra 2

Series and Summation NotationSeries and Summation Notation

Holt Algebra 2

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Lesson QuizLesson Quiz



Series and Summation Notation

Warm UpFind the first 5 terms of each sequence.

1. 4.

2.

5.

3.



Warm Up ContinuedWrite a possible explicit rule for the nth term of each sequence.

6. 1, 2, 4, 8, 16, …

7. 4, 7, 10, 13, 16, …



Evaluate the sum of a series expressed in sigma notation.

Objective



seriespartial sumsummation notation

Vocabulary



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 asequence. Some examples are shown in the table.



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.





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.



Write the series in summation notation.

Example 1A: Using 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




Example 1B: Using Summation Notation


Explicit formula.

Summation notation.




Caution!



Check It Out! Example 1a

Write each series in summation notation.


Explicit formula.


Summation notation.



Check It Out! Example 1b



Explicit formula.


Summation notation.



Expand the series and evaluate.

Example 2A: Evaluating a Series

Expand the series by replacing k.

Evaluate powers.

Simplify.



Expand the series and evaluate.

Example 2B: Evaluating a Series

Expand.

Simplify.

= (12 – 10) + (22 – 10) + (32 – 10) + (42 – 10) + (52 – 10) + (62 – 10)

= –9 – 6 – 1 + 6 + 15 + 26

= 31




Expand each series and evaluate.

= (2(1) – 1) + (2(2) – 1) + (2(3) – 1) + (2(4) – 1)

= 1 + 3 + 5 + 7

= 16


Simplify.




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)


Simplify.



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.



The formula for the sum of a constant series

is as shown.



A linear series is a counting series, such as the sum of the first 10 natural numbers.

Examine when the terms are rearranged.



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





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



Evaluate the series.

Example 3A: Using Summation Formulas

Method 1 Use the summation formula.

There are 7 terms.

Method 2 Expand and evaluate.

Constant series



Example 3B: Using Summation Formulas



Linear series




Example 3C: Using Summation Formulas


Method 2 Use a graphing calculator.

Quadratic series







Constant series

There are 60 terms.

= nc = 60(4) = 240


= 60 + 60 + 60 + 60

4 items

= 240



Linear series


Evaluate each series.


= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 120




Check It Out! Example 3c


= 385

Quadratic series


Method 2 Use a graphing calculator.

n(n + 1)(2n + 1) 6=

10(10 + 1)(2 · 10 + 1) 6

=

(110)(21) 6=



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?

Example 4: Problem-Solving Application



11 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.



22 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.



Solve33

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.



Solve33

Evaluate the series for several n-values.

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(11)

=

2(1) + 2(2) + 2(3) + 2(4) + 2(5) + 2(6) + 2(7) + 2(8) + 2(9) + 2(10)

=

= 110

= 132



Solve33

Because Sam has only 144 stones, the patio can have at most 11 complete rows.

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



Look Back44

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.



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?



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.



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.



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.



Look Back44

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



Lesson Quiz: Part I


1. 1 – 10 + 100 – 1000 + 10,000

2.


3.

4.

5.

6.

55

64

325

285



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 tohave as many rows as possible with 100 games. How many rows will Ann’s display have?5

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