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Chapter 11Sequences, Series, and the Binomial Theorem
§ 11.1
Sequences and Summation Notation
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 11.1
Sequences
Many creations in nature involve intricate mathematical designs, including a variety of spirals. For example, the arrangement of the individual florets in the head of a sunflower forms spirals. In some species, there are 21 spirals in the clockwise direction and 34 in the counterclockwise direction. The precise numbers depend on the species of sunflower: 21 and 34, or 34 and 55, or 55 and 89, or 89 and 144.
This observation is interesting because these numbers correspond to a special sequence of numbers in mathematics:
1,1,2,3,5,8,13,21,34,55,89,144,233,…
Can you find the pattern? Maybe so… but if not, here it is: The first two numbers are 1, and every term thereafter is the sum of the two preceding terms. This special sequence is called the Fibonacci sequence of numbers after Leonardo of Pisa, also known as Fibonacci, who was an Italian mathematician of the thirteenth century.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 11.1
Sequences
Definition of a SequenceAn infinite sequence is a function whose domain is the set of positive integers. The function values, or terms, of the sequence are represented by
Sequences whose domains consist only of the first n positive integers are called finite sequences.
,,,,, 321 naaaa
na
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 11.1
Sequences
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Write the first four terms of the sequence whose general term is given.
We need to find the first four terms of the sequence whose general
term is To do so, we replace n in the formula with 1, 2, 3, and 4.
12
1- 1
n
n
na
.
12
1- 1
n
n
na
3
1
12
1
12
1- 2
1
11
1
a
5
1
14
1
12
1- 3
2
12
2
a
9
1
18
1
12
1- 4
3
13
3
a
17
1
116
1
12
1- 5
4
14
4
a
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 11.1
Sequences - Factorials
Factorial NotationIf n is a positive integer, the notation n! (read “n factorial”) is the product of all positive integers from n down through 1.
n! = n(n – 1)(n – 2) (3)(2)(1)
0! (zero factorial), by definition, is 1.
0! = 1
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 11.1
Sequences - Factorials
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Write the first four terms of the sequence whose general term is given.
We need to find the first four terms of the sequence. To do so, we replace each n in the formula with 1, 2, 3, and 4.
2
! 1
n
nan
2
1
2
1
! 2
1
! 1121
a
2
3
4
6
4
3!
2
! 1222
a
3
8
9
24
9
! 4
3
! 1323
a
2
15
16
120
16
5!
4
! 1424
a
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 11.1
Sequences - Summations
Summation NotationThe sum of the first n terms of a sequence is represented by the summation notation
where i is the index of summation, 1 is the lower limit of summation, and n is the upper limit of summation.
,1
321 n
n
ii aaaaa
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 11.1
Sequences - Summations
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Expand and evaluate the sum:
We must replace i in the expression with all
consecutive integers from 0 to 4, inclusive. Then we add.
.
! 1
14
0
1
i
i
i
! 1
1 1
i
i
! 14
1
! 13
1
! 12
1
! 11
1
! 10
1
! 1
1 14131211104
0
1
i
i
i
! 5
1
! 4
1
! 3
1
! 2
1
! 1
1 54321
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 11.1
Sequences - Summations
120
1
24
1
6
1
2
1
1
1
CONTINUECONTINUEDD
120
1
24
1
6
1
2
11
120
1
5
5
24
1
20
20
6
1
60
60
2
1
120
1201
120
1
120
5
120
20
120
60
120
120
30
19
120
76
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 11.1
Sequences - Summations
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Express the sum using summation notation:
We will use 1 as the lower limit of summation and i for the index of summation.
.216
16
5
3
4
2
3
1
The sum
216
16
5
3
4
2
3
1
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 11.1
Sequences - Summations
has 16 terms, each of the form , starting at i = 1 and ending at i = 16. Thus, 2i
iCONTINUECONTINUE
DD
16
1
.2216
16
5
3
4
2
3
1
i i
i
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 11.1
Sequences in Application
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
A deposit of $10,000 is made in an account that earns 8% interest compounded quarterly. The balance in the account after n quarters is given by the sequence
Since every year has four quarters (of a year), and interest is being compounded for six years, interest will be compounded for 4 x 6 = 24 quarters. Therefore, we will replace n in the given equation with 24.
,3,2,1,4
08.01000,10
na
n
n
Find the balance in the account after six years. Round to the nearest cent.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 11.1
Sequences in Application
Therefore, the balance in the account after six years is $16,084.37.
24
24 4
08.01000,10
a
CONTINUECONTINUEDD
2402.01000,10
2402.1000,10
608437.1000,10
37.084,16
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 11.1
Sequences
What is a sequence anyway?
A sequence is just a list of numbers having a pattern. You can think of a sequence as a function. The terms of the sequence are the range values, and the set of positive integers is the domain.
For the Fibonacci series, this would be represented by:
f(1) =1, f(2)=1,f(3)=2, f(4)=3,f(5)=5,f(6)=8,f(7)=13 and so on.
Some sequences are finite and others are infinite.
