9.2 Arithmetic Sequences

Objective

• To find specified terms and the common difference in an arithmetic sequence.

• To find the partial sum of a arithmetic series.

• A sequence whose consecutive terms have a common difference is call an arithmetic sequence.

Definition of an ArithmeticSequence

• A sequence is arithmetic if the differences between consecutive terms are the same.

• So the sequenceis arithmetice if there

is a number d such that

• The number d is the common difference of the arithmetic sequence.

1 2 3, , ,... ,...na a a a

2 1 3 2 4 3 ...a a a a a a d

Example 1

• a). The sequence whose nth term is 4n + 3 is arithmetic.

• b). The sequence whose nth term is 7 – 5n is arithmetic.

• c). The sequence whose nth term is 1

( 3)4

n

• Notice that each of the arithmetic sequences has an nth term that is of the form dn + c, whose common difference of the sequence is d.

• An arithmetic sequence may be thought of as a linear function whose domain is the set of natural numbers.

The nth Term of an Arithmetic Sequence

• The nth term of an arithmetic sequence has the form

where d is the common difference between consecutive terms of the sequence and

na dn c

1c a d

Example 2Finding the nth term of an

Arithmetic Sequence• Find a formula for

the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2.

1

1

2, 3

2 3 1

3 1n

a d

c a d

c

a n

Example 3Writing the Terms of an

Arithmetic Sequence• The fourth term

of an arithmetic sequence is 20 and the 13th term is 65. Write the first several terms of this sequence.

4

13

20 (4)

65 (13)

two equations in 2 unknowns

subtracting 45 9

5, 20 5(4) , 0

5

5,10,15,20,25n

a d c

a d c

d

d c c

a n

Example 4Using a recursive Formula

• Find the ninth term of the arithmetic sequence that begins with 2 and 9.

1

1

9

2, 7

, 2 7 5

7 5

7(9) 5 58n

a d

c a d c

a n

a

The Sum of a Finite Arithmetic Sequence

• The sum of a finite arithmetic sequence with n terms is

1( )2n n

nS a a

Example 5Finding the Sum of a Finite

Arithmetic Sequence

• Find the sum 1 + 3 + 5 + 7 + 9 + 11 + 13 +15 + 17 + 19

1

1 20

20

20

( )2

10, 1, 19

10(1 19)

25(20)

n n

nS a a

n a a

S

S

Example 6Finding the Sum of a Finite

Arithmetic Sequence• Find the sum of the integers (a)

from 1 to 100 and (b) from 1 to N.

1

100

100

100

100, 1, 100

100(1 100)

250(101)

5050

nn a a

S

S

S

1

100

2

100

, 1,

(1 )2

2 2

nn n a a n

nS n

n nS

Partial Sum

• The sum of the first n terms of an infinite sequence is the nth partial sum

Example 7

• Find the 150th partial sum of the arithmetic sequence.

• 5, 16, 27, 38, 49, . . .

1

1

150

150

150

150

150, 5, 11

5 11 6

5 6

5(150) 6 1644

150(5 1644)

275(1649)

123675

n

n a d

c a d

a n

a

S

S

S

Example 8Seating Capacity

• An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?

1

1

20

20

20

20

20, 20, 1

20 1 19

19

20 19 39

20(20 39)

210(59)

590

n

n a d

c a d

a n

a

S

S

S

Example 9Total Sales

• A small business sells $10,000 worth of products during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years this business is in operation.

1

1

10

10

20

20

10, $10,000, 7500

10,000 7500 2500

7500 2500

7500(10) 2500 77,500

10(10,000 77,500)

25(87500)

437500

n

n a d

c a d

a n

a

S

S

S