Digital Lesson
Arithmetic Sequences
and Series
An infinite sequence is a function whose domain
is the set of positive integers.
a1, a2, a3, a4, . . . , an, . . .
terms
The first three terms of the sequence an = 4n – 7 are
a1 = 4(1) – 7 = – 3
a2 = 4(2) – 7 = 1
finite sequence
a3 = 4(3) – 7 = 5.
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A sequence is arithmetic if the differences
between consecutive terms are the same.
4, 9, 14, 19, 24, . . .
arithmetic sequence
9–4=5
14 – 9 = 5
19 – 14 = 5
The common difference, d, is 5.
24 – 19 = 5
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Example: Find the first five terms of the sequence
and determine if it is arithmetic.
an = 1 + (n – 1)4
a1 = 1 + (1 – 1)4 = 1 + 0 = 1
a2 = 1 + (2 – 1)4 = 1 + 4 = 5
a3 = 1 + (3 – 1)4 = 1 + 8 = 9
d=4
a4 = 1 + (4 – 1)4 = 1 + 12 = 13
a5 = 1 + (5 – 1)4 = 1 + 16 = 17
This is an arithmetic sequence.
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The nth term of an arithmetic sequence has the
form
an = dn + c
where d is the common difference and c = a1 – d.
a1 = 2
2, 8, 14, 20, 26, . . . .
c=2–6=–4
d=8–2=6
The nth term is 6n – 4.
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Example: Find the formula for the nth term of an
arithmetic sequence whose common difference is 4
and whose first term is 15. Find the first five terms
of the sequence.
an = dn + c
a1 – d = 15 – 4 = 11
= 4n + 11
The first five terms are
a1 = 15
15, 19, 23, 27, 31.
d=4
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The sum of a finite arithmetic sequence with n
terms is given by
S n  n (a1  an).
2
5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ?
n = 10
a1 = 5
a10 = 50
Sn  10 (5  50)  5(55)  275
2
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The sum of the first n terms of an infinite sequence
is called the nth partial sum.
Sn  n (a1  an)
2
Example: Find the 50th partial sum of the arithmetic
sequence – 6, – 2, 2, 6, . . .
a1 = – 6
d=4
an = dn + c = 4n – 10
c = a1 – d = – 10
a50 = 4(50) – 10 = 190
Sn  50 (6  190)  25(184)  4600
2
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Graphing Utility: Find the first 5 terms of the arithmetic
sequence an = 4n + 11.
beginning
variable
value
end value
List Menu:
100
Graphing Utility: Find the sum   2n .
i 1
List Menu:
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lower limit
upper
limit
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The sum of the first n terms of a sequence is
represented by summation notation.
upper limit of summation
n
a  a  a
i 1
i
1
2
 a3  a4 
 an
lower limit of summation
index of
summation
5
 1  n   (1 1)  (1 2)  (1 3)  (1 4)  (1 5)
i 1
 2  3 4  5  6
 20
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Example: Find the partial sum.
100
  2n   2(1)  2(2)  2(3) 
 246
i 1
a1
 2(100)
 200
a100
S100  n (a1  a100)  100 (2  200)
2
2
 50(202)  10,100
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Consider the infinite sequence a1, a2, a3, . . ., ai, . . ..
1. The sum of the first n terms of the sequence is called
a finite series or the partial sum of the sequence.
n
a1 + a2 + a3 + . . . + an   ai
i 1
2. The sum of all the terms of the infinite sequence is
called an infinite series.

a1 + a2 + a3 + . . . + ai + . . .   ai
i 1
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

i
Example: Find the fourth partial sum of  5 1 .
2
i 1
4
        
 5 1   5 1   5 1   5 1 
2
4
8
16
i
1
2
3
1  5 1 5 1 5 1 5 1
5

2
2
2
2
2
i1
4
 555 5
2 4 8 16
 40  20  10  5  75
16 16 16 16 16
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