Arithmetic Sequences

A mathematical model for the average annual salaries of major league baseball players generates the following data.

1,438,0001,347,0001,256,0001,165,0001,074,000983,000892,00

0801,000Salary

19981997199619951994199319921991Year

From 1991 to 1992, salaries increased by $892,000 - $801,000 = $91,000. From 1992 to 1993, salaries increased by $983,000 - $892,000 = $91,000. If we make these computations for each year, we find that the yearly salary increase is $91,000. The sequence of annual salaries shows that each term after the first, 801,000, differs from the preceding term by a constant amount, namely 91,000. The sequence of annual salaries

801,000, 892,000, 983,000, 1,074,000, 1,165,000, 1,256,000....

is an example of an arithmetic sequence.

Arithmetic Sequences

Definition of an Arithmetic Sequence

• An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence.

The recursion formula an an 1 24 models the thousands of Air Force personnel on active duty for each year starting with 1986. In 1986, there were 624 thousand personnel on active duty. Find the first five terms of the arithmetic sequence in which a1 624 and an an 1 24.

Solution The recursion formula an an 1 24 indicates that each term after the first is obtained by adding 24 to the previous term. Thus, each year there are 24 thousand fewer personnel on active duty in the Air Force than in the previous year.

Text Example

The recursion formula an an 1 24 models the thousands of Air Force personnel on active duty for each year starting with 1986. In 1986, there were 624 thousand personnel on active duty. Find the first five terms of the arithmetic sequence in which a1 624 and an an 1 24.

The first five terms are

624, 600, 576, 552, and 528.

Solution

a1 624 This is given. a2 a1 – 24 624 – 24 600 Use an an 1 24 with n 2.

a3 a2 – 24 600 – 24 576 Use an an 1 24 with n 3.

a4 a3 – 24 576 – 24 552 Use an an 1 24 with n 4.

a5 a4 – 24 552 – 24 528 Use an an 1 24 with n 5.

Text Example cont.

Example• Write the first six terms of the arithmetic sequence where

a1 = 50 and d = 22

•a1 = 50 a2 = 72 a3 = 94 a4 = 116 a5 = 138 a6 = 160

Solution:

General Term of an Arithmetic Sequence

• The nth term (the general term) of an arithmetic sequence with first term a1 and common difference d is

• an = a1 + (n-1)d

Find the eighth term of the arithmetic sequence whose first term is 4 and whose common difference is 7.

Solution To find the eighth term, as, we replace n in the formula with 8, a1 with 4, and d with 7.

an a1 (n 1)d

a8(8 1)(7) 4 ) 4 (49) 5

The eighth term is 45. We can check this result by writing the first eight terms of the sequence:

4, 3, 10, 17, 24, 31, 38, 45.

Text Example

The Sum of the First n Terms of an Arithmetic Sequence

• The sum, Sn, of the first n terms of an arithmetic sequence is given by

in which a1 is the first term and an is the nth term.

)(2 1 nn aan

S

Example

690)69(10

)636(10

)3*196(6(10

)(2

2020120

aaS

Solution:

• Find the sum of the first 20 terms of the arithmetic sequence: 6, 9, 12, 15, ...

Example

15

1

)42(i

i

18012*15

)24(2

15

))26()2((2

15

)42(15

1

i

i

• Find the indicated sum

Solution:

Arithmetic Sequences