1. The numbers in sequences are called terms.
USING AND WRITING SEQUENCES
The numbers in sequences are called terms.
You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.

2. USING AND WRITING SEQUENCES
DOMAIN:
The domain gives the relative position of each term.
The range gives the terms of the sequence.
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.
Writing Terms of Sequences
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
a 3 = 2(3) + 3 = 9
3rd term
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. f (1) = (–2) 1 – 1 = 1 f (2) = (–2) 2 – 1 = –2 f (3) = (–2) 3 – 1 = 4
Writing Terms of Sequences
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,
Writing Rules for Sequences
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 __
– , , – , , …. 1 3 9 27 81 _

6. Writing Rules for Sequences
SOLUTION n 1
243 - 5 1 3  , 9 27 81
terms
rewrite
terms 1 3 - 4 , 2 1 3 - 5 1 3 -
A rule for the nth term is an = n

7. n Describe the pattern, write the next term, and write
Writing Rules for Sequences
Describe the pattern, write the next term, and write
a rule for the n th term of the sequence.
SOLUTION
2, 6, 12 , 20,…. n 5 30
terms
rewrite
terms
1(1 +1)
2(2 +1)
3(3 +1)
4(4 +1)
5(5 +1)
A rule for the nth term is f (n) = n (n+1).

8. You can graph a sequence by letting the horizontal axis
Graphing a Sequence
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. • Write a rule for the number of oranges in each layer.
Graphing a Sequence
You work in the produce department of a grocery store and 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. From the diagram, you can see that an = n 2
Graphing a Sequence
SOLUTION
The diagram below shows the first three layers of the stack. Let an represent the number of oranges in layer n. n 1 2 3 an
1 = 1 2
4 = 2 2
9 = 3 2
From the diagram, you can see that an = n 2

11. an = n2 Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100).
Graphing a Sequence
an = n2
Plot the points (1, 1), (2, 4),
(3, 9), , (10, 100).

12. USING SERIES
When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite.
FINITE SEQUENCE
FINITE SERIES
3, 6, 9, 12, 15
INFINITE SEQUENCE
INFINITE SERIES
3, 6, 9, 12, 15, . . .
. . .
You can use summation notation to write a series. For example, for the finite series shown above, you can write
=  3i 5
i = 1

13.  3i 5 i = 1 3 + 6 + 9 + 12 + 15 =  3i USING SERIES
upper limit of summation
Is read as
“the sum from i equals 1 to 5 of 3i.” 5
i = 1  3i
=  3i 5
i = 1
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 similar to that for a finite series. For example, for the infinite series shown earlier, you can write:
=  3i 
i = 1
. . .
The infinity symbol,  , indicates that the series continues without end.

15. The index of summation does not have to be i.
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. Writing Series with Summation Notation
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 terms of the series can be written as:
ai = 5i where i = 1, 2, 3, , 20
The summation notation is
 5i. 20
i = 1

17. Writing Series with Summation Notation
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 series can be written as:
ai = where i = 1, 2, 3, i
i + 1 
The summation notation for the series is 
i = 1 i
i + 1 .

18. Writing Series with Summation Notation
The sum of the terms of a finite sequence can be found by simply adding the terms. For sequences with many terms, however, adding the terms can be tedious. Formulas for finding the sum of the terms of three special types of sequences are shown next.

19.  1 = n  i =  i 2 = n (n + 1) n (n + 1)(2 n + 1) n 2 6
Writing Series with Summation Notation
CONCEPT
SUMMARY
FORMULAS FOR SPECIAL SERIES n
i = 1
 1 = n
gives the sum of n 1’s . 1
 i =
n (n + 1) 2 n
i = 1
gives the sum of positive integers from 1 to n . 2
 i 2 =
n (n + 1)(2 n + 1) 6 n
i = 1
gives the sum of squares of positive integers from 1 to n. 3

20. Using a Formula for a Sum
RETAIL DISPLAYS How many oranges are in a square pyramid 10 layers high?

21. Using a Formula for a Sum
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 = 1
i 2 =
. . . = 6
10(10 + 1)(2 • )
10(11)(21) = 6 =
There are 385 oranges in the stack.