Holt Algebra 2 12-2 Series and Summation Notation 12-2 Series and Summation Notation Holt Algebra 2...
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Transcript of Holt Algebra 2 12-2 Series and Summation Notation 12-2 Series and Summation Notation Holt Algebra 2...
Holt Algebra 2
12-2Series and Summation Notation12-2 Series and Summation Notation
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 2
12-2Series and Summation Notation
Warm UpFind the first 5 terms of each sequence.
1. 4.
2.
5.
3.
Holt Algebra 2
12-2Series and Summation Notation
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, …
Holt Algebra 2
12-2Series and Summation Notation
Evaluate the sum of a series expressed in sigma notation.
Objective
Holt Algebra 2
12-2Series and Summation Notation
seriespartial sumsummation notation
Vocabulary
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
Write the series in summation notation.
Example 1B: Using Summation Notation
Find a rule for the kth term of the sequence.
Explicit formula.
Summation notation.
Write the notation for the first 6 terms.
Holt Algebra 2
12-2Series and Summation Notation
Caution!
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
Expand the series and evaluate.
Example 2A: Evaluating a Series
Expand the series by replacing k.
Evaluate powers.
Simplify.
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
The formula for the sum of a constant series
is as shown.
Holt Algebra 2
12-2Series and Summation Notation
A linear series is a counting series, such as the sum of the first 10 natural numbers.
Examine when the terms are rearranged.
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
Example 3B: Using Summation Formulas
Method 1 Use the summation formula.
Method 2 Expand and evaluate.
Linear series
Evaluate the series.
Holt Algebra 2
12-2Series and Summation Notation
Example 3C: Using Summation Formulas
Method 1 Use the summation formula.
Method 2 Use a graphing calculator.
Quadratic series
Evaluate the series.
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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=
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
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
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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?
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
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.
Holt Algebra 2
12-2Series and Summation Notation
Look Back44
Use the diagram to continue the pattern. The 6th loop would be 294 inches. S6 = 34 + 40 + 46 + 52 + 58 + 64 = 294.
Holt Algebra 2
12-2Series and Summation Notation
Lesson Quiz: Part I
Write each series in summation notation.
1. 1 – 10 + 100 – 1000 + 10,000
2.
Write each series in summation notation.
3.
4.
5.
6.
55
64
325
285
Holt Algebra 2
12-2Series and Summation Notation
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