Arithmetic Sequences

How do I define an arithmetic sequence and how do I use the formula to find different terms of

the sequence?

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

Arithmetic Sequences and Series

Arithmetic Sequence: sequence whose consecutive terms have a common difference.

Example: 3, 5, 7, 9, 11, 13, ...

The terms have a common difference of 2.

The common difference is the number d.

To find the common difference you use an+1 – an

Example: Is the sequence arithmetic? –45, –30, –15, 0, 15, 30

Yes, the common difference is 15How do you find any term in this sequence?To find any term in an arithmetic sequence, use the

formula an = a1 + (n – 1)dwhere d is the common difference.

The first term of an arithmetic sequence is . We add d to get the next term. There is a pattern, therefore there is a formula we can use to give use any term that we need without listing the whole sequence .

1a

The nth term of an arithmetic sequence is given by:

1 ( 1)na a n d The last # in the sequence/or the # you are looking for

First term

The position the term is in The common

difference

Find the 14th term of the arithmetic sequence4, 7, 10, 13,……

1 ( 1)na a n d 14a (14 1) 4 3

4 (13)3 4 39 43

1 ( 1)na a n d

In the arithmetic sequence 4,7,10,13,…, which term has a value of 301?

301 4 ( 1)3n 301 4 3 3n 301 1 3n 300 3n100 n

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

n 1

n 1 n

nth term of arithmetic sequence

sum of n terms of arithmetic sequen

a a n 1 d

nS a a

2ce

Given an arithmetic sequence with 15 1a 38 and d 3, find a .

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

x

15

38

NA

-3

n 1a a n 1 d

38 x 1 15 3

X = 80

1 29Find d if a 6 and a 20

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-6

29

20

NA

x

n 1a a n 1 d

120 6 29 x

26 28x

13x

14

n 1Find n if a 633, a 9, and d 24

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

9

x

633

NA

24

n 1a a n 1 d

633 9 21x 4

633 9 2 244x

X = 27

16 1Find a if a 1.5 and d 0.5 Try this one:

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

1.5

16

x

NA

0.5

n 1a a n 1 d

16 1.5 0.a 16 51

16a 9

Example: Find a formula for the nth term of the arithmetic sequence in which the common difference

is 5 and the first term is 3.

an = a1 + (n – 1)d a1 = 3 d = 5an = 3 + (n – 1)5

Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence?

an = a1 + (n – 1)d We need to determine what the first term is...

d = 4 and a5 = 15a5 = a1 + (5 – 1)4 = 15a1 = –1

a10 = –1 + (10 – 1)4 a10 = 35

An arithmetic mean of two numbers, a and b, is simply their average. Using the arithmetic mean we can also form a sequence.

Insert three arithmetic means between 8 and 16.

Let 8 be the 1st termLet 16 be the 5th term 1 ( 1)na a n d

16 8 (5 1)d 2d

8, , , ,1610 12 14

Find two arithmetic means between –4 and 5

-4, ____, ____, 5

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-4

4

5

NA

x

n 1a a n 1 d

15 4 4 x x 3

The two arithmetic means are –1 and 2, since –4, -1, 2, 5

forms an arithmetic sequence

Find three arithmetic means between 1 and 4

1, ____, ____, ____, 4

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

1

5

4

NA

x

n 1a a n 1 d

4 1 x15 3

x4

The three arithmetic means are 7/4, 10/4, and 13/4

since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence

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.

An arithmetic series is a series associated with an arithmetic sequence.

The sum of the first n terms:

12n n

nS a a

1(2 ( 1) )2n

nS a n d

Find the sum of the first 100 natural numbers.

1 + 2 + 3 + 4 + … + 100

12n n

nS a a 1 1a

100na 100n

100

100(1 100)

2S

5050

Find the sum of the first 14 terms of the arithmetic series 2 + 5 + 8 + 11 + 14 + 17 +…

1 2a 3d 14n

1(2 ( 1) )2n

nS a n d

14

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

2S

14 7(3 13(3))S 7(43)301