The numbers in sequences are called terms'

1
The numbers in sequences are called terms.
You can think of a sequence as a function whose
domainis a set of consecutive integers. If a
domain is notspecified, it is understood that
the domain starts with 1.
2
1 2 3 4 5
DOMAIN
The domain gives the relative position of each
term.
The range gives the terms of the sequence.
3 6 9 12 15
RANGE
This is a finite sequence having the rule
an 3n,
where an represents the nth term of the
sequence.
3
Write the first six terms of the sequence an 2n
3.
SOLUTION
a 1 2(1) 3 5
1st term
a 2 2(2) 3 7
2nd term
3rd term
a 3 2(3) 3 9
a 4 2(4) 3 11
4th term
a 5 2(5) 3 13
5th term
a 6 2(6) 3 15
6th term
4
Write the first six terms of the sequence f (n)
(2) n 1 .
SOLUTION
f (1) (2) 1 1 1
1st term
f (2) (2) 2 1 2
2nd term
f (3) (2) 3 1 4
3rd term
f (4) (2) 4 1 8
4th term
f (5) (2) 5 1 16
5th term
f (6) (2) 6 1 32
6th term
5
If the terms of a sequence have a recognizable
pattern, then you may be able to write a rule
for the n th term of the sequence.
Describe the pattern, write the next term, and
write a rule for the n th term of the sequence
6
SOLUTION
1 2 3 4
7
Describe the pattern, write the next term, and
write a rule for the n th term of the sequence.
SOLUTION
2, 6, 12 , 20,.
1 2 3 4
2 6 12 20
5(5 1)
A rule for the nth term is f (n) n (n1).
8
You can graph a sequence by letting the
horizontal axis represent the position numbers
(the domain) and the vertical axis represent the
terms (the range).
9
You work in the producedepartment of a grocery
storeand are stacking oranges in the shape of
square pyramid with ten layers.
Write a rule for the number of oranges in
each layer.
Graph the sequence.
10
SOLUTION
The diagram below shows the first three layers of
the stack. Let an represent the number of
oranges in layer n.
From the diagram, you can see that an n 2
11
Plot the points (1, 1), (2, 4), (3, 9), . . . ,
(10, 100).
12
USING SERIES
When the terms of a sequence are added, the
resultingexpression is a series. A series can
be finite or infinite.
. . .
You can use summation notation to write a
series. Forexample, for the finite series shown
above, you can write
13
USING SERIES
Is read as the sum from i equals 1 to 5 of 3i.
upper limit of summation
index of summation
lower limit of summation
14
USING SERIES
Summation notation is also called sigma notation
because it uses the uppercase Greek letter
sigma, written ?.
Summation notation for an infinite series is
similarto that for a finite series. For example,
for the infiniteseries shown earlier, you can
write
The infinity symbol, 8, indicates that the series
continues without end.
15
USING SERIES
The index of summation does not have to be I.
Any letter can be used. Also, the index does not
have to begin at 1.
16
Example Write the series represented by the
summation notation . Then find the sum.

• 12 12 12 12 0! 1! 2!
  3!
• 12 12 12 12 1 1 2
  6
• 32

17
Write the series with summation notation.
SOLUTION
Notice that the first term is 5 (1), the second
is 5 (2),the third is 5 (3), and the last is 5
(20). So the termsof the series can be written
as
ai 5i where i 1, 2, 3, . . . , 20
18
Write the series with summation notation.
SOLUTION
Notice that for each term the denominator of the
fraction is 1 more than the numerator. So, the
terms of the seriescan be written as
19
The sum of the terms of a finite sequence can be
foundby simply adding the terms. For sequences
with manyterms, however, adding the terms can be
tedious. Formulas for finding the sum of the
terms of three special types of sequences are
shown next.
20
gives the sum of n 1s .
gives the sum of positive integers from 1 to n .
gives the sum of squares of positive integers
from 1 to n.
21
RETAIL DISPLAYS How many oranges are in a
square pyramid 10 layers high?
22
SOLUTION
You know from the earlier example that the i th
term of the series is given by ai i 2, where i
1, 2, 3, . . . , 10.
10
. . .
?
i 2 12 22 102
i 1
385
There are 385 oranges in the stack.