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.

The domain gives the relative position of each term.

1 2 3 4 5 DOMAIN:

3 6 9 12 15RANGE:The range gives the terms of the sequence.

This is a finite sequence having the rule

an = 3n,where an represents the nth term of the sequence.

USING AND WRITING SEQUENCES

n

an

Writing Terms of Sequences

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

SOLUTION

a 1 = 2(1) + 3 = 5 1st term

2nd term

3rd term

4th term

6th term

a 2 = 2(2) + 3 = 7

a 3 = 2(3) + 3 = 9

a 4 = 2(4) + 3 = 11

a 5 = 2(5) + 3 = 13

a 6 = 2(6) + 3 = 15

5th 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

2nd term

3rd term

4th term

6th term

f (2) = (–2) 2 – 1 = –2

f (3) = (–2) 3 – 1 = 4

f (4) = (–2) 4 – 1 = – 8

f (5) = (–2) 5 – 1 = 16

f (6) = (–2) 6 – 1 = – 32

5th term

Writing Rules for Sequences

If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the n th term of the sequence.

Describe the pattern, write the next term, and write a rule for the n th term of the sequence

____ – , , – , , ….13

19

127

181

_ _

1 3

, 1 9

, 1 27

, 1 81

Writing Rules for Sequences

SOLUTION

1 2 3 4n

terms 1243

5

13

4

13

1, 1

3

2, 1

3

3, 1

3

5rewriteterms

13

A rule for the nth term is an =

n

2 6 12 20

Writing Rules for Sequences

SOLUTION

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

terms

5(5 +1)

Describe the pattern, write the next term, and write a rule for the n th term of the sequence.

2, 6, 12 , 20,….

5

30

1 2 3 4

rewriteterms 1(1 +1) 2(2 +1) 3(3 +1) 4(4 +1)

n

Graphing a Sequence

You can graph a sequence by letting the horizontal axisrepresent 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

SOLUTION

The diagram below shows the first three layers of the

stack. Let an represent the number of oranges in layer n.

n 1 2 3

an 1 = 1 2 4 = 2 2 9 = 3

2

From the diagram, you can see that an = n

2

Graphing a Sequence

Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100).

an = n2

USING SERIES

. . .

3 + 6 + 9 + 12 + 15 = ∑ 3i5

i = 1

FINITE SEQUENCE

FINITE SERIES

3, 6, 9, 12, 15

3 + 6 + 9 + 12 + 15

INFINITE SEQUENCE

INFINITE SERIES

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

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

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

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

3 + 6 + 9 + 12 + 15 = ∑ 3i5

i = 1

USING SERIES

5

i = 1∑3i

Is read as “the sum from i equals 1 to 5 of 3i.”

index of summation lower limit of summation

upper limit of summation

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:

3 + 6 + 9 + 12 + 15 + = ∑ 3i∞

i = 1. . .

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.

Example: Write the series represented by the summation notation . Then find the sum.

= 12 + 12 + 12 + 12 0! 1! 2! 3!

= 12 + 12 + 12 + 12 1 1 2 6

= 32

12n!k=0

3

∑

Writing Series with Summation Notation

Write the series with summation notation.

5 + 10 + 15 + + 100. . .

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

The summation notation is ∑ 5i.20

i = 1

Writing Series with Summation Notation

Notice that for each term the denominator of the fractionis 1 more than the numerator. So, the terms of the seriescan be written as:

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

Write the series with summation notation.

SOLUTION

. . .1 2 3 42 3 4 5

+ + + +

∞The summation notation for the series is ∑

i = 1

ii + 1

.

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

FORMULAS FOR SPECIAL SERIESCONCEPT

SUMMARY

n

i = 1∑ 1 = n

∑ i = n (n + 1)

2

n

i = 1

1

2

3

gives the sum of positive integers from 1 to n .

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

∑ i 2 = n (n + 1)(2 n + 1)

6

n

i = 1

gives the sum of n 1’s .

Using a Formula for a Sum

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

Using a Formula for a Sum

SOLUTION

You know from the earlier example that the i th term of the series is given by ai = i

2, where i = 1, 2, 3, . . . , 10.

∑10

i = 1i

2 = 12+ 22 + + 102 . . .

10(11)(21)=

6

= 385

There are 385 oranges in the stack.

=6

10(10 + 1)(2 • 10 + 1)