U SING AND W RITING S EQUENCES

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U SING AND W RITING S EQUENCES. The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1. U SING AND W RITING S EQUENCES. n. a n.

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USING AND WRITING SEQUENCES

The numbers in sequences are called terms.

You can think of a sequence as a function whose domainis a set of consecutive integers. If a domain is notspecified, it is understood that the domain starts with 1.

USING AND WRITING SEQUENCES

n

an

1 2 3 4 5

DOMAIN:

The domain gives the relative positionof each term.

The range gives the terms of the sequence.

3 6 9 12 15

RANGE:

This is a finite sequence having the rule

an= 3n,

where anrepresents the nth term of the sequence.

Writing Terms of Sequences

Write the first six terms of the sequence an = 2n + 3.

SOLUTION

a1= 2(1) + 3 = 5

1st term

a2= 2(2) + 3 = 7

2nd term

a3= 2(3) + 3 = 9

3rd term

a4= 2(4) + 3 = 11

4th term

a5= 2(5) + 3 = 13

5th term

a6= 2(6) + 3 = 15

6th term

Writing Terms of Sequences

Write the first six terms of the sequence f(n) = (–2)n – 1 .

SOLUTION

f(1) = (–2)1 – 1 = 1

1st term

f(2) = (–2)2 – 1 = –2

2nd term

f(3) = (–2)3 – 1 = 4

3rd term

f(4) = (–2)4 – 1 = – 8

4th term

f(5) = (–2)5 – 1 = 16

5th term

f(6) = (–2)6 – 1 = – 32

6th term

Writing Rules for Sequences

1

1

1

1

_

_

__

__

– , , – , , ….

3

9

27

81

If the terms of a sequence have a recognizable pattern,

then you may be able to write a rule for the nth term

of the sequence.

Describe the pattern, write the next term, and write

a rule for the nth term of the sequence

Writing Rules for Sequences

5

n

1

243

1

3

1

9

1

27

1

81

-

terms

,

,

,

1

2

3

4

5

rewrite

terms

1

3

1

3

1

3

1

3

1

3

,

,

,

-

-

-

-

-

n

1

3

A rule for the nth term is an =

-

SOLUTION

12 3 4

Writing Rules for Sequences

n

5

terms

30

rewrite

terms

1(1 +1)

2(2 +1)

3(3 +1)

4(4 +1)

Describe the pattern, write the next term, and write

a rule for the nth term of the sequence.

2, 6, 12 , 20,….

SOLUTION

12 3 4

2 6 12 20

5(5 +1)

A rule for the nth term is f (n) = n(n+1).

Graphing a Sequence

You can graph a sequence by letting the horizontal axis

represent the position numbers (the domain) and the vertical axis represent the terms (the range).

Graphing a Sequence

You work in the producedepartment of a grocery storeand are stacking oranges in the shape of square pyramid with ten layers.

• Write a rule for the number of oranges in each layer.

• Graph the sequence.

Graphing a Sequence

n

1

2

3

4 = 22

an

1 = 12

9 = 32

SOLUTION

The diagram below shows the first three layers of the stack. Let an represent the number of oranges in layer n.

From the diagram, you can see that an= n2

Graphing a Sequence

an= n2

Plot the points (1, 1), (2, 4),

(3, 9), . . . , (10, 100).

FINITE SEQUENCE

INFINITE SEQUENCE

3, 6, 9, 12, 15

3, 6, 9, 12, 15, . . .

FINITE SERIES

INFINITE SERIES

3 + 6 + 9 + 12 + 15

3 + 6 + 9 + 12 + 15 + . . .

5

3 + 6 + 9 + 12 + 15 = 3i

i = 1

USING SERIES

When the terms of a sequence are added, the resultingexpression is a series. A series can be finite or infinite.

. . .

You can use summation notation to write a series. Forexample, for the finite series shown above, you can write

5

3i

i= 1

5

3 + 6 + 9 + 12 + 15 = 3i

i = 1

USING SERIES

“the sum from iequals 1 to 5 of 3i.”

upper limit of summation

index of summation

lower limit of summation

3 + 6 + 9 + 12 + 15 + = 3i

. . .

i = 1

USING SERIES

Summation notation is also called sigma notation

because it uses the uppercase Greek letter sigma,

written .

Summation notation for an infinite series is similarto that for a finite series. For example, for the infiniteseries shown earlier, you can write:

The infinity symbol,  , indicates that the series continues without end.

USING SERIES

The index of summation does not have to be i.

Any letter can be used. Also, the index does not

have to begin at 1.

Writing Series with Summation Notation

. . .

5 + 10 + 15 + + 100

20

5i.

The summation notation is

i = 1

Write the series with summation notation.

SOLUTION

Notice that the first term is 5(1), the second is 5(2),the third is 5(3), and the last is 5(20). So the termsof the series can be written as:

ai= 5i where i = 1, 2, 3, . . . , 20

Writing Series with Summation Notation

i

ai= where i = 1, 2, 3, 4 . . .

i + 1

1 2 3 4

. . .

+ + + +

i

2 3 4 5

The summation notation for the series is

.

i +1

i= 1

Write the series with summation notation.

SOLUTION

Notice that for each term the denominator of the fraction

is 1 more than the numerator. So, the terms of the seriescan be written as:

Writing Series with Summation Notation

The sum of the terms of a finite sequence can be foundby simply adding the terms. For sequences with manyterms, however, adding the terms can be tedious. Formulas for finding the sum of the terms of three special types of sequences are shown next.

Writing Series with Summation Notation

CONCEPT

FORMULAS FOR SPECIAL SERIES

SUMMARY

n

1 = n

i = 1

3

1

2

n

n(n + 1)

i =

2

i = 1

n

n(n + 1)(2n + 1)

i2 =

6

i = 1

gives the sum of n1’s.

gives the sum of positive integers from 1 to n.

gives the sum of squares of positive integers from 1 to n.

Using a Formula for a Sum

RETAIL DISPLAYSHow many oranges are in a square pyramid 10 layers high?

Using a Formula for a Sum

10

. . .

i2 = 12+ 22 + + 102

i = 1

10(10 + 1)(2 • 10 + 1)

=

6

10(11)(21)

=

6

SOLUTION

You know from the earlier example that the ith term of the series is given by ai= i2, where i = 1, 2, 3, . . . , 10.

= 385

There are 385 oranges in the stack.