9.1 Sequences and Series

Recursive SequenceFactorial Notation

Summation Notation

Definition of a Sequence

The values of a function whose domain is positive set of integers. f(1), f(2), f(3), f(4),…f(n)....called terms written as

a1, a2, a3, a4, …, an, …

This is an infinite sequence since it does not end. If the sequence has a final term, it is called a finite sequence.

Find the first 4 terms of the sequence an = 2n + 1

n = 1 a1 = 2(1) + 1 = 3

n = 2 a2= 2(2) + 1 = 5

n = 3 a3 = 2(3) + 1 = 7

n = 4 a4 = 2(4) + 1 = 9

3, 5, 7, 9

Find the first 4 terms of the sequence

n = 1

n = 2

n = 3

n = 4

12

1 1

na

n

n

1

11

1121 11

1

a

3

1122

1 12

2

a

5

1132

1 13

3

a

7

1142

1 14

4

a

71,

51,

31,1

Recursive Sequence

A sequence where each term of the sequence is defined as a function of the preceding terms.

Example: an = an -2 + an-1 : a1 = 1, a2 = 1

a3 = 1 + 1 = 2

a4 = 1 + 2 = 3

a5 = 2 + 3 = 5

1,1, 2, 3, 5, …….

The Fibonacci Sequence is a Recursive Sequence

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:

By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

Why are they called FibonacciLeonardo Pisano Bigollo (c. 1170 – c. 1250) also known

as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."

Fibonacci is best known to the modern world for the spreading of the Hindu-Arabic numeral system in Europe,

Definition of Factorial Notation

If n is a positive integer, n factorial is

0! = 1

nnn 14321!

12054321!5

Find the first 5 terms of the sequence

Start where n = 1

!3n

an

n

Find the first 5 terms of the sequence

Start where n = 1

!3n

an

n

4081

120243

!53

827

2481

!43

29

627

!33

29

!23

3!13

5

5

4

4

3

3

2

2

1

1

a

a

a

a

a

Evaluate

Expand

32187654321654321

!3!8!6

Evaluate

Expand

32187654321654321

!3!8!6

3361

321871

!3!8!6

Summation Notation or Sigma Notation

If you add the terms of a sequence, the sequence is called a Series.

bfafnfb

an

.......Index variable

Upper limit

Lower limit

Series

Find each Sum

•

3

1

4n

n

Find each Sum

•

241284

34241443

1

n

n

Find each Sum

•

6

3

2 2n

n

Find each Sum

•

783423147

262524232 22226

3

2

n

n

Properties of Sums

If c is a constant

n

kk

n

kk

n

i

acca

cnc

11

1

Properties of Sums

n

k

n

kk

n

kkkk

n

k

n

k

n

kkkkk

baba

baba

1 11

1 1 1

Find the Sum

4

1

62n

n

Find the Sum

4

1

4

1

4

1

6262nnn

nn

Find the Sum

44248642

4642322212

62624

1

4

1

4

1

nnn

nn

The Infinite Series

Infinite Series go on forever, some converge on one number, other decrease or increase forever.

Does this Series ever end in one number?

4232221221n

n

The Infinite Series

Infinite Series go on forever, some converge on one number, another decrease or increase forever.

Does this Series ever end in one number?

10241

2561

641

161

41

41

1nn

Homework

Page 621 – 622 # 1, 10, 17, 28,

40, 55, 58, 66, 68, 74, 77, 84,

87, 92