Pc Intro To Seq And Series

Writin g a nd Usin g Seq uen ce s MATH

PRO JECT

USING AND WRITING SEQUENCES

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.

USING AND WRITING SEQUENCES DOMAIN: n

an RANGE:

1

3

2

6

3

9

4

12

5

15

The domain gives  the relative position  of each term.

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.

Writing Terms of Sequences

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

a 1 = 2(1) + 3 = 5

1st  term

a 2 = 2(2) + 3 = 7

2nd  term

a 3 = 2(3) + 3 = 9

3rd  term

a 4 = 2(4) + 3 = 11

4th  term

a 5 = 2(5) + 3 = 13

5th  term

a 6 = 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

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 1 , __ 1 , …. – _1 , _1 , – __ 3 9 27 81

Writing Rules for Sequences

SOLUTION

n

terms

rewrite terms

−

1

2

1 , 3

1 , − 1 9

1

1 , − 3

3

2

1 , − 3

4

,

2 7 3 1 , − 3

5

1 81

1 − 3

− 1 243

4

1 − A rule for the nth term is an = 3

1 − 3 n

5

Writing Rules for Sequences

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

n terms rewrite terms

2, 6, 12 , 20,…. 1

2

3

4

5

2

6

12

20

30

1(1 +1)

2(2 +1)

3(3 +1)

4(4 +1)

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 produce department of a grocery store and 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

an

1 = 12

2

4=22

3

9 = 32

From the diagram, you can see that an = n

2

Graphing a Sequence

an = n2 Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100).

USING SERIES When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite. 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 + . . . .

You can use summation notation to write a series. For example, for the finite series shown above, you can write 5

3 + 6 + 9 + 12 + 15 = Σ 3i i=1

USING SERIES upper limit of  summation

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

5

Σ 3i

5

3 + 6 + 9 + 12 + 15 = Σ 3i

i=1 index of summation

i=1

lower limit of summation

USING SERIES

Summation notation is also called sigma notation because it uses the uppercase Greek letter sigma, written Σ. Summation notation for an infinite series is similar to that for a finite series. For example, for the infinite series shown earlier, you can write: ∞

3 + 6 + 9 + 12 + 15 + . . . = Σ 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.

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 terms of the series can be written as:

ai = 5i where i = 1, 2, 3, . . . , 20 The summation notation is

20

Σ 5i.

i=1

Writing Series with Summation Notation

Write the series with summation notation. 1 + 2 + 3 + 4 +... 2 3 4 5 SOLUTION Notice that for each term the denominator of the fraction is 1 more than the numerator. So, the terms of the series can be written as:

ai =

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

The summation notation for the series is

∞

i . i=1i+1

Σ

Writing Series with Summation Notation

The sum of the terms of a finite sequence can be found by simply adding the terms. For sequences with many terms, 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 SUMMARY

1

2

n

Σ

gives the sum of n 1’s .

1 =n

 

i=1 n

Σ

i=

i=1

3

FORMULAS FOR SPECIAL SERIES

n

Σ

i

i=1

2=

n (n + 1) 2

gives the sum of positive  integers from 1 to n .

n (n + 1)(2 n + 1) gives the sum of squares  6

of positive integers from  1 to n.

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

Σ i 2 = 12+ 22 + . . . + 102

i=1

= = =

10(10 + 1)(2 • 10 + 1) 6 10(11)(21) 6 385

There are 385 oranges in the stack.