678 CHAPTER 12 Sequences and Series
OBJECTIVES In the Palace of the Alhambra, residence of the
Moorish rulers of Granada, Spain, the Sultana’s
quarters feature an interesting architectural
pattern:
There are 2 matched marble slabs inlaid in
the floor, 4 walls, an octagon (8-sided) ceiling,
16 windows, 32 arches, and so on.
If this pattern is continued indefinitely, the set
of numbers forms an infinite sequencewhose
termsare powers of 2.
Sequences and Series
12.1
1 Find the terms of a
sequence, given the
general term.
2 Find the general
term of a sequence.
3 Use sequences to
solve applied
problems.
4 Use summation
notation to
evaluate a series.
5 Write a series with
summation
notation.
6 Find the arithmetic
mean (average) of a
group of numbers.SequenceAn infinite sequenceis a function with the set of all positive integers as the do-
main. A finite sequenceis a function with domain of the form
where nis a positive integer.
5 1, 2, 3,Á, n 6 ,
OBJECTIVE 1 Find the terms of a sequence, given the general term.
For any positive integer n, the function value of a sequence is written as (read
“asub-n”). The function values written in order, are the termsof the
sequence, with the first term, the second term, and so on. The expression
which defines the sequence, is called the general termof the sequence.
In the Palace of the Alhambra example, the first five terms of the sequence are
and
The general term for this sequence is
Writing the Terms of Sequences from the General TermGiven an infinite sequence with find the following.
(a)The second term of the sequence
Replace nwith 2.(b) (c)
NOW TRYGraphing calculators can be used to generate and graph sequences, as shown in
FIGURE 1on the next page. The calculator must be in dot mode, so that the discrete
points on the graph are not connected. Remember that the domain of a sequence
consists only of positive integers.
a 12 = 12 + 12 =^14412
1
12
=
145
12
a 10 = 10 + 10 =^10010
1
10
=
101
10
a 2 = 2 +
1
2
=
5
2
an=n+^1 n ,
EXAMPLE 1
an= 2 n.
a 1 =2, a 2 = 4, a 3 = 8, a 4 = 16, a 5 = 32.
a 1 a 2 an,
a 1 , a 2 , a 3 ,Á,
an
NOW TRY
EXERCISE 1
Given an infinite sequence
with find .an= 5 - 3 n, a 3
NOW TRY ANSWER
- a 3 =- 4