copyright © 2011 pearson education, inc. slide 11.1-1 obj: the student will demonstrate the ability...

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

Obj: The student will demonstrate the ability to evaluate the first five terms of explicit and recursive sequences.

DrillWhat is the next shape/number for each?1.

2.5, 3, 1, -1, -3, ____3.1, 4, 9, 16, 25, ____4.2, 4, 8, 16, 32, ____


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


n

n

na2

1

Replacing n with n = 1, 2, 3, 4, and 5 will give you the first five terms.


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.

n 1 2 3 4 5 6

an 1 2.66 6.24 10.4 9.11 10.2

n 7 8 9 10

an 9.31 10.1 9.43 9.98


Class workName:____________

Write the first five terms of each sequence.(explicit)1. an = 2n + 3 2. an = n3 + 1

1. an = 3(2n ) 4. an = (-1)n (n)

Find the third, fourth and fifth terms of each.(recursive)3. a1 = 6; an = an-1 + 4 6. a1 = 1; an = an-1 + 2n – 1

7. a1 = 9; an = an-1 8. a1 = 4; an = (an-1 )2 - 10

3

1


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


Examples of Finite Series

1.

2.

3.

6

2k

k

7

3

74n

n

5

1

4k

k



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 obj: the student will demonstrate the ability...

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