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Chapter 11 Sec 1
Arithmetic Sequences
Algebra 2 Chapter 11 Sections 1 & 2
What is a Sequence
• A sequence is a list of numbers in a
particular order.
• Each number in a sequence is called a term.
• The first term is symbolized by a1, the
second term, a2 and so on.
• An arithmetic sequence is found by adding a
constant called the common difference.
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Algebra 2 Chapter 11 Sections 1 & 2
Example 1
• Find the next four terms of the arithmetic sequence
55, 49, 43, … .
Find the common difference d by subtracting two consecutive terms.
49 – 55 = –6 and 43 – 49 = –6 , So d = –6
Now add –6 to the third term of the sequence, and then
continue adding – 6 …
55, 49, 43,
37
–6
31
–6
25
–6
19
–6
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Algebra 2 Chapter 11 Sections 1 & 2
The nth term of an Arithmetic Sequence
Write an equation for the nth term of 8, 17, 26, 35, …
In this sequence a1 = 8 and d = 9, an = a1 +(n – 1)d
an = 8 + (n – 1)9
an = 8 + 9n – 9 = 9n – 1
The equation for the nth term is an = 9n – 1
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Algebra 2 Chapter 11 Sections 1 & 2
Arithmetic Means
Sometimes you are given two terms and are asked to find
the terms between, these terms between are called
arithmetic means.
Find the three arithmetic means between 21 and 45.
33 ___,
39 45
27 ___,
21, ___,
a1 a2 a3
a4 a5
an = a1 + (n – 1)d
a5 = 21 + (5 – 1)d
45 = 21 + 4d
24 = 4d
6 = d So…
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Chapter 11 Sec 2
Arithmetic Series
Algebra 2 Chapter 11 Sections 1 & 2
Series
• The first amphitheaters were built for contests
between gladiators.
• Modern amphitheaters are usually used for
performing arts.
• Amphitheaters generally get wider as the distance
from the stage increases.
• Suppose a small amphitheater can seat 18 in the
first row and each row can seat 4 more people
than the previous row.
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Algebra 2 Chapter 11 Sections 1 & 2
Arithmetic Series
• The number of seats in the rows of the
amphitheater form an arithmetic sequence. To find
the number of people who could sit in the first
four rows, add the first four terms of the
sequence. That sum is 18 + 22 + 26 + 30 or 96.
• A series is an indicated sums of the terms of a
sequence.
• Since 18, 22, 26, 30 is an arithmetic sequence, 18
+ 22 + 26 + 30 is an arithmetic series.
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Algebra 2 Chapter 11 Sections 1 & 2
Arithmetic Series
Arithmetic Sequence Arithmetic Series.
5, 8, 11, 14, 17
5 + 8 + 11 + 14 + 17
–9, –3, 3
–9 + (–3) + 3
Sn represents the sum of the first n terms of a series. For
example, S4 is the sum of the first four terms.
An arithmetic sequence, Sn has n terms and the sum of the
first and last terms is a1 + an .
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Algebra 2 Chapter 11 Sections 1 & 2
Example 1
Find the sum of the first 100 positive integers.
The series is 1 + 2 + … + 100.
a1 = 1, a100 = 100 and d = 1
n
S n  a1  an 
2
100
1  100
Sn 
2
Sn  50101  5050
n
S n  2a1  n  1d 
2
100
21  100  11
Sn 
2
Sn  50101  5050
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Algebra 2 Chapter 11 Sections 1 & 2
Example 2
Find the first three terms of an arithmetic series in
which a1 = 9, an = 105 and Sn = 741.
Since you know a1, an, and Sn find n and use that to find d.
n
an  a1  n  1d
S n  a1  an 
2
105  9  13 1d
n
741  9  105
2
96  12d
741  57n
8d
n  13
9, 17, 25
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Algebra 2 Chapter 11 Sections 1 & 2
Sigma Notation
Writing out a series can be time-consuming and lengthy.
For convenience, there is a more concise notation called
sigma notation.
The series 3 + 6 + 9 + … + 30 can be
10
expressed as
 3n
n 1
Last value of n
n = top – bottom +1
10
3
n

Formula for the
terms of the series
n 1
First value of n
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Algebra 2 Chapter 11 Sections 1 & 2
Sigma Notation
10
Evaluate
 2k  1
k 3
a1  23  1  7
an  210  1  21
n 8
n
S n  a1  an 
2
8
S8  7  21
2
S8  428  112
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Algebra 2 Chapter 11 Sections 1 & 2
Daily Assignment
•
Chapter 11 Sections 1 & 2
•
Study Guide (SG)
Pg 141 & 143 All
• Pg 142 & 144 Even
•
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Algebra 2 Chapter 11 Sections 1 & 2
3, 4, 5, 6, 7, 8, 9, 10
a1 a2 a3 a4 a5 a6 a7 a8
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