Transcript A2CH12L2

12-2
12-2Series
Seriesand
andSummation
SummationNotation
Notation
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
12-2 Series and Summation Notation
Warm Up
Find the first 5 terms of each sequence.
1.
4.
2.
5.
3.
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12-2 Series and Summation Notation
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, …
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12-2 Series and Summation Notation
Objective
Evaluate the sum of a series expressed
in sigma notation.
Holt Algebra 2
12-2 Series and Summation Notation
Vocabulary
series
partial sum
summation notation
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12-2 Series 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 a
sequence. Some examples are shown in the table.
Holt Algebra 2
12-2 Series 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.
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12-2 Series and Summation Notation
Holt Algebra 2
12-2 Series 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.
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12-2 Series and Summation Notation
Example 1A: Using Summation Notation
Write the series in 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
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12-2 Series and Summation Notation
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.
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12-2 Series and Summation Notation
Caution!
Holt Algebra 2
12-2 Series 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
12-2 Series 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.
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12-2 Series and Summation Notation
Example 2A: Evaluating a Series
Expand the series and evaluate.
Expand the series
by replacing k.
Evaluate powers.
Simplify.
Holt Algebra 2
12-2 Series and Summation Notation
Example 2B: Evaluating a Series
Expand the series and evaluate.
= (12 – 10) + (22 – 10) + (32 – 10) +
(42 – 10) + (52 – 10) + (62 – 10)
= –9 – 6 – 1 + 6 + 15 + 26
= 31
Holt Algebra 2
Expand.
Simplify.
12-2 Series and Summation Notation
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)
=1+3+5+7
= 16
Holt Algebra 2
Simplify.
12-2 Series and Summation Notation
Check It Out! Example 2b
Expand each series and evaluate.
Expand the series by replacing k.
= –5(2)(1 – 1) – 5(2)(2
5(2)(5 – 1)
– 1)
– 5(2)(3
– 1)
= – 5 – 10 – 20 – 40 – 80 Simplify.
= –155
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– 5(2)(4
– 1)
–
12-2 Series 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.
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12-2 Series and Summation Notation
The formula for the sum of a constant series
is
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as shown.
12-2 Series 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
12-2 Series and Summation Notation
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
Similar methods will help you find the sum of a
quadratic series.
Holt Algebra 2
12-2 Series and Summation Notation
Holt Algebra 2
12-2 Series and Summation Notation
Caution
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
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12-2 Series and Summation Notation
Example 3A: Using Summation Formulas
Evaluate the series.
Constant series
Method 1 Use the
summation formula.
There are 7 terms.
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Method 2 Expand and
evaluate.
12-2 Series and Summation Notation
Example 3B: Using Summation Formulas
Evaluate the series.
Linear series
Method 1 Use the
summation formula.
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Method 2 Expand and
evaluate.
12-2 Series and Summation Notation
Example 3C: Using Summation Formulas
Evaluate the series.
Quadratic series
Method 1 Use the
summation formula.
Holt Algebra 2
Method 2 Use a
graphing calculator.
12-2 Series and Summation Notation
Check It Out! Example 3a
Evaluate the series.
Constant series
Method 1 Use the
summation formula.
There are 60 terms.
Method 2 Expand and
evaluate.
= 60 + 60 + 60 + 60
4 items
= nc = 60(4) = 240
= 240
Holt Algebra 2
12-2 Series and Summation Notation
Check It Out! Example 3b
Evaluate each series.
Linear series
Method 1 Use the
summation formula.
Method 2 Expand and
evaluate.
=1+2+3+4+5
+6+7+8+9+
10 + 11 + 12 + 13
+ 14 + 15 = 120
Holt Algebra 2
12-2 Series and Summation Notation
Check It Out! Example 3c
Evaluate the series.
Quadratic series
Method 1 Use the
summation formula.
n(n + 1)(2n + 1)
=
6
= 10(10 + 1)(2 · 10 + 1)
6
(110)(21)
=
6
= 385
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Method 2 Use a
graphing calculator.
12-2 Series and Summation Notation
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?
Holt Algebra 2
12-2 Series and Summation Notation
1
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.
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12-2 Series and Summation Notation
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.
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12-2 Series and Summation Notation
3
Solve
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.
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12-2 Series and Summation Notation
3
Solve
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
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12-2 Series and Summation Notation
3
Solve
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.
Holt Algebra 2
12-2 Series and Summation Notation
4
Look Back
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.
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12-2 Series 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
12-2 Series and Summation Notation
1
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
12-2 Series and Summation Notation
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.
Holt Algebra 2
12-2 Series and Summation Notation
3
Solve
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
12-2 Series and Summation Notation
4
Look Back
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
12-2 Series 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.
55
3.
4.
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64
5.
325
6.
285
12-2 Series 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 to
have as many rows as possible with 100 games.
How many rows will Ann’s display have?
5
Holt Algebra 2