Copyright © 2011 Pearson Education, Inc. Slide 11.1-1.


Chapter 11: Further Topics in Algebra

11.1 Sequences and Series

11.2 Arithmetic Sequences and Series

11.3 Geometric Sequences and Series

11.4 Counting Theory

11.5 The Binomial Theorem

11.6 Mathematical Induction

11.7 Probability


11.1 Sequences

Sequences are ordered lists generated by a

function, for example f(n) = 100n

(1), (2), (3),...

100,200,300,...

f f f


• f (x) notation is not used for sequences.• Write • Sequences are written as ordered lists

• a1 is the first element, a2 the second element, and so on

11.1 Sequences

A sequence is a function that has a set of natural numbers (positive integers) as its domain.

( )na f n

1 2 3, , , ...a a a


11.1 Graphing Sequences

The graph of a sequence, an, is the graph of thediscrete points (n, an) for n = 1, 2, 3, …

Example Graph the sequence an = 2n.

Solution


11.1 Sequences

A sequence is often specified by giving a formula forthe general term or nth term, an.

Example Find the first four terms for the sequence

Solution

1

2n

na

n

1 2(1 1) /(1 2) 2 / 3, (2 1) /(2 2) 3 / 4a a

3 4(3 1) /(3 2) 4 / 5, (4 1) /(4 2) 5 / 6a a


11.1 Sequences

• A finite sequence has domain the finite set

{1, 2, 3, …, n} for some natural number n.

Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

• An infinite sequence has domain

{1, 2, 3, …}, the set of all natural numbers.

Example 1, 2, 4, 8, 16, 32, …


11.1 Convergent and Divergent Sequences

• A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number.

• A sequence that is not convergent is said to be divergent.



Example The sequence converges to 0.

The terms of the sequence 1, 0.5, 0.33.., 0.25, … grow smaller and smaller approaching 0. This can be seen graphically.

1na

n



Example The sequence is divergent.

The terms grow large without bound

1, 4, 9, 16, 25, 36, 49, 64, …

and do not approach any one number.

2na n


11.1 Sequences and Recursion Formulas

• A recursion formula or recursive definition defines a sequence by– Specifying the first few terms of the sequence

– Using a formula to specify subsequent terms in terms of preceding terms.


11.1 Using a Recursion Formula

Example Find the first four terms of the sequence a1 = 4; for n >1, an = 2an-1 + 1

Solution We know a1 = 4.

Since an = 2an-1 + 1

2 1

3 2

4 3

2 1 2 4 1 9

2 1 2 9 1 19

2 1 2 19 1 39

a a

a a

a a


11.1 Applications of Sequences

Example The winter moth population in thousandsper acre in year n, is modeled by

for n > 2

(a) Give a table of values for n = 1, 2, 3, …, 10

(b) Graph the sequence.

21 1 11, 2.85 0.19n n na a a a


11.1 Applications of Sequences

Solution(a)

(b)Note the population stabilizes near a value of 9.7 thousand insects per acre.

654321n

10.29.1110.46.242.661an

10987n

9.989.4310.19.31an


11.1 Series and Summation Notation

• Sn is the sum a1 + a2 + …+ an of the first n terms of the sequence a1, a2, a3, … .

is the Greek letter sigma and indicates a sum.

• The sigma notation means add the terms ai

beginning with the 1st term and ending with the nth term.

• i is called the index of summation.

1

n

ii

a



A finite series is an expression of the form

and an infinite series is an expression of the form

.

1 2 31

...n

n n ii

S a a a a a

1 2 31

... ...n ii

S a a a a a



Example Evaluate

(a) (b)

Solution(a)

(b)

6

1

(2 1)k

k

6

3j

j

a

61 2 3 4

1

5 6

(2 1) (2 1) (2 1) (2 1) (2 1)

(2 1) (2 1)

3 5 9 17 33 65 132

k

k

6

3 4 5 63

jj

a a a a a



Summation Properties

If a1, a2, a3, …, an and b1, b2, b3, …, bn are two sequences, and c is a constant, then, for every positive integer n,

(a) (b)

(c)

1

n

i

c nc

1 1

n n

i ii i

ca c a

1 1 1

( )n n n

i i i ii i i

a b a b



Summation Rules

1

2 2 2 2

1

2 23 3 3 3

1

( 1)1 2 ...

2

( 1)(2 1)1 2 ...

6

( 1)1 2 ...

4

n

i

n

i

n

i

n ni n

n n ni n

n ni n



Example Use the summation properties to

evaluate (a) (b) (c)

Solution

(a)

40

1

5i

22

1

2i

i

142

1

(2 3)i

i

40

1

5 40(5) 200i



Solution

(b)

(c)

14 14 14 14 142 2 2

1 1 1 1 1

(2 3) 2 3 2 3

14(14 1)(2 14 1)2 14(3) 1988

6

i i i i i

i i i

22 22

1 1

22(22 1)2 2 2 506

2i i

i i

Copyright © 2011 Pearson Education, Inc. Slide 11.1-1.

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