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Arithmetic Arithmetic Sequences Sequences and Seriesand Series
An An Arithmetic SequenceArithmetic Sequence is is defineddefined as a sequence in as a sequence in which there is a which there is a common common
differencedifference between between consecutive terms.consecutive terms.
Which of the following sequences are arithmetic? Identify the common
difference.
3, 1, 1, 3, 5, 7, 9, . . .
15.5, 14, 12.5, 11, 9.5, 8, . . .
84, 80, 74, 66, 56, 44, . . .
8, 6, 4, 2, 0, . . .
50, 44, 38, 32, 26, . . .
YES 2d
YES
YES
NO
NO
1.5d
6d
The common
difference is
always the
difference between
any term and the
term that proceeds
that term.26, 21, 16, 11, 6, . . .
Common Difference = 5
The general form of an ARITHMETIC sequence.
1aFirst Term:
Second Term: 2 1a a d
Third Term:
Fourth Term:
Fifth Term:
3 1 2a a d
4 1 3a a d
5 1 4a a d
nth Term: 1 1na a n d
Formula for the nth term of an ARITHMETIC sequence.
1 1na a n d
The nth termna
The term numbern
The common differenced
1 The 1st terma
If we know any
If we know any three of these we
three of these we ought to be able
ought to be able to find the fourth.
to find the fourth.
Given: 79, 75, 71, 67, 63, . . .Find: 32a
1 79
4
32
a
d
n
1
32
32
1
79 32 1 4
45
na a n d
a
a
IDENTIFY SOLVE
Given: 79, 75, 71, 67, 63, . . .
Find: What term number is -169?
1 79
4
169n
a
d
a
1 1
169 79 1 4
63
na a n d
n
n
IDENTIFY SOLVE
Given:10
12
3.25
4.25
a
a
1
3
3.25
4.25
3
a
a
n
1 1
4.25 3.25 3 1
0.5
na a n d
d
d
IDENTIFY SOLVE
Find: 1a
What’s the real question? The Difference
Given:10
12
3.25
4.25
a
a
10 3.25
0.5
10
a
d
n
1
1
1
1
3.25 10 1 0.5
1.25
na a n d
a
a
IDENTIFY SOLVE
Find: 1a
Arithmetic Arithmetic SeriesSeries
Write the first three terms and the
Write the first three terms and the last two terms of the following
last two terms of the following arithmetic series.arithmetic series.
50
1
73 2p
p
71 69 67 . . . 25 27
What is the sum of What is the sum of
this series?this series?
71 69 67 . . . 25 27
27 25 . . . 67 69 71
44 44 44 . . . 44 44 44
50 71 27
2
110071 + (-27) Each sum is the same.
50 Terms
1 1 1 12 . . . 1a a d a d a n d
1 1 1 11 . . . 2a n d a d a d a
1
2nn a as
1
Sum
Number of Terms
First Term
Last Termn
S
n
a
a
1 1 1 1 1 11 1 . . . 1a a n d a a n d a a n d
Find the sum of the terms of this arithmetic series.
35
1
29 3k
k
1
2nn a a
S
1
35
35
26
76
n
a
a
35 26 76
2875
S
S
Find the sum of the terms of this arithmetic series. 151 147 143 139 . . . 5
1
2nn a a
S
1
40
40
151
5
n
a
a
40 151 5
22920
S
S
1 1
5 151 1 4
40
na a n d
n
n
What term is -5?What term is -5?
Alternate formula for the
sum of an Arithmetic
Series.
1
2nn a
Sa
1 1Substitute na a n d
1 1
1
1
2
2 1
2
n a a n dS
n a n dS
1
# of Terms
1st Term
Difference
n
a
d
Find the sum of this series 36
0
2.25 0.75j
j
2.25 3 3.73 4.5 . . .
12 1
2
n a n dS
It is not convenient to It is not convenient to find the last term.find the last term.
1
37
2.25
0.75
n
a
d
37 2 2.25 37 1 0.75
2582.75
S
S
35
1
45 5i
i
1
2nn a a
S
12 1
2
n a n dS
135 40 130nn a a 135 40 5n a d
35 40 130
21575
S
S
35 2 40 35 1 3
21575
S
S
