Arithmetic Sequences

Definition of an arithmetic sequence.

An arithmetic sequence is a sequence in which each term butthe first is found by adding a constant, called the common difference d, to the previous term.

Example 1. The table below showsthe cost of mailing a first class letter in 1995.

Ounces

Cost

1

.32

2

.55

3

.78

4

1.01

5

1.24

Find how much it costs to mail letters that weigh 6,7, and 8 oz.

Example 1. How much to mail letters that weigh 6,7, and 8 oz.First find the common difference.

Cost .32 .55 .78 1.01 1.24

+.23 +.23 +.23 +.23

.23 or 23 cents is the common difference.

Example 1. The table below showsthe cost of mailing a first class letter in 1995. d = .23

Ounces

Cost

1

.32

2

.55

3

.78

4

1.01

5

1.24

Ounces

Cost

6

1.47

7

1.70

8

1.93

Example 2. Find the next fourterms of the arithmetic sequence91, 83, 75, ....

91 83 75

-8 -8

The common difference is -8.

Example 2. Find the next fourterms of the arithmetic sequence91, 83, 75, ....

91 83 75

-8 -8

The next four terms are

67 59 51 43

There is a pattern in the way theterms of an arithmetic sequenceare formed.

It is possible to develop a formula that expresses each term of anarithmetic sequence in terms of thefirst term a1 and the common difference d.

Let’s look at example 2.

numerical

symbols

91 83 75 67

a1 a2 a3 a4 an

In terms of d where d = -8

a1 = 91+0(-8) a2 = 91+1(-8)

a3 = 91+2(-8) a4 = 91+3(-8)

Let’s look at example 2.numericalsymbols

91 83 75 67a1 a2 a3 a4 an

In terms of d a1 = 91+0(-8) a2 = 91+1(-8)

a3 = 91+2(-8) a4 = 91+3(-8)Therefore an = 91+(n-1)(-8)

Equivalently an = a1+(n-1)(-8)

Formula for the nth term of an arithmetic sequence.

The nth term of an arithmetic sequence with first term a1 and common difference d is given by

where n is a positive integer.

an = a1+(n-1)(d)

Example 3.A radio station is giving away atA radio station is giving away atleast $1000.00 in a contest. For least $1000.00 in a contest. For each caller who answers the each caller who answers the question incorrectly the station question incorrectly the station adds $97.00 to the jackpot. If you adds $97.00 to the jackpot. If you are the 18th caller and the first toare the 18th caller and the first toanswer correctly how much do youanswer correctly how much do youwin?win?

Example 3. Radio station contest.This is an arithmetic sequence withThis is an arithmetic sequence withaa11 = 1000 and d = 97. = 1000 and d = 97.

aann = a = a11 + (n-1)d + (n-1)d

aa1818 = 1000 + (18-1)(97) = 1000 + (18-1)(97)

aa1818 = 1000 + 17(97) = 1000 + 17(97)

aa1818 = 1000 + 1649 = 1000 + 1649 = 2649= 2649

The terms between any two nonconsecutive terms of an arithmetic sequence are called thearithmetic means.

In the sequence14, 23, 32, 41, 50, 59, 68, 77

32, 41, and 50 are the three arithmetic means between 23 and 59

Example 4.

Find the four arithmetic means between 18 and 78.

Use the nth term formula to find d.

18, ____, ____, ____, ___, 78

18 is a1 78 is a6

Example 4.Find the four arithmetic means between 18 and 78.Use the nth term formula to find d.18, ____, ____, ____, ___, 78

18 is a1 78 is a6

a6 = a1 + 5(d)

78 = 18 + 5(d) 12 = d

Example 4.Find the four arithmetic means between 18 and 78.

18, ____, ____, ____, ___, 78

12 = d Now use d to find the termsa2 = 18 + 1(12)

a3 = 18 + 2(12)

a4 = 18 + 3(12)

a5 = 18 + 4(12)

Terms are 30, 42, 54, and 66

Example 5.

Write an equation for the nth termof the arithmetic sequence 6, 13, 20, 27. ...

In this sequence a1 = 6 and d = 7

Therefore an = 6 + (n-1)7

an = 6 + 7n - 7 an = 7n - 1