ARITHMETIC & GEOMETRIC SEQUENCES Section 14.2

ARITHMETIC SEQUENCES

An arithmetic sequence is a sequence in which each term differs from the proceeding term by a constant amount d. The constant d is called the common difference of the sequence.

Ex. 2, 6, 10, 14, 18,…

first term: common difference = 4

EXAMPLE 1:

Write the first five terms of the arithmetic sequence whose first term is 7 and whose common difference is 2.

The first five terms are 7, 9, 11, 13, 15.

NOTICE THE GENERAL FORM?

GENERAL TERM OF AN ARITHMETIC SEQUENCE The general term of an arithmetic sequence

is given by

where is the first term and d is the common difference.

EXAMPLE 2:

Consider the arithmetic sequence whose first term is 3 and common difference is .

a. Write an expression for the general term .

Plug in and d

Distribute the -5

Combine like terms

EXAMPLE 2:

Consider the arithmetic sequence whose first term is 3 and common difference is .

b. Find the twentieth term of this sequence.

Plug in 20 for n

Simplify

OYO:

Consider the arithmetic sequence whose first term is 2 and whose common difference is .

a. Write an expression for the general term .

b. Find the twelfth term of the sequence.

EXAMPLE 3:

Find the eleventh term of the arithmetic sequence whose first three terms are 2, 9, and 16.

What do we know?

Arithmetic – so it must have a common difference, d

+7

+7

So the eleventh term is 72.

OYO:

Find the ninth term of the arithmetic sequence whose first three terms are 3, 9, and 15.

51

EXAMPLE 4:

If the third term of an arithmetic sequence is 12 and the eighth term is 27, find the fifth term.

What do we know?

How many terms are we jumping?

8 – 3 = 5

What is the difference between these terms?

27 – 12 = 15

Divide these to get d:15 / 5 = 3

EXAMPLE 4:

If the third term of an arithmetic sequence is 12 and the eighth term is 27, find the fifth term.

What do we know?

d = 3

EXAMPLE 4:

If the third term of an arithmetic sequence is 12 and the eighth term is 27, find the fifth term.

What do we know?d = 3

Finding the fifth term:

12

OYO:

If the third term of an arithmetic sequence is 23 and the eighth term is 63, find the sixth term.

47

EXAMPLE 5: APPLICATION Donna had an offer for a job starting at $40,000 per year

and guaranteeing her a raise of $1600 a year for the next 5 years. Write the general term for the arithmetic sequence that models Donna’s potential annual salaries, and find her salary for the fourth year.

What do we know?

EXAMPLE 5: APPLICATION Donna had an offer for a job starting at $40,000 per year

and guaranteeing her a raise of $1600 a year for the next 5 years. Write the general term for the arithmetic sequence that models Donna’s potential annual salaries, and find her salary for the fourth year.

6400

$44,800

GEOMETRIC SEQUENCES

A geometric sequence is a sequence in which each term is obtained by multiplying the preceding term by a constant r. The constant r is called the common ratio of the sequence.

Ex. 12, 6, 3, 3/2, …

First term: Common ratio:

EXAMPLE 6:

Write the first five terms of a geometric sequence whose first term is 7 and whose common ratio is 2.

The first five terms are 7, 14, 28, 56, 112.

142856

GENERAL TERM OF A GEOMETRIC SEQUENCE

The general term of a geometric sequence is given by

Where is the first term and r is the common ratio.

EXAMPLE 7:

Find the eighth term of the geometric sequence whose first term is 12 and whose common ratio is .

OYO:

Find the seventh term of the geometric sequence whose first term is 64 and whose common ratio is

EXAMPLE 8:

Find the fifth term of the geometric sequence whose first three terms are 2, -6, and 18.

What do we know?

Geometric – so it must have a common ratio, r

∗−3

So the fifth term is 162.

∗−3

OYO:

Find the fifth term of the geometric sequence whose first three terms are -3, 6, and -12.

516∙45=¿

EXAMPLE 9:

If the second term of a geometric sequence is and the third term is , find the first term and the common ratio.

1

1

1

4

14

𝑟=14 ¿

54∙41=5

OYO:

If the second term of a geometric sequence is and the third term is , find the first term and the common ratio.

EXAMPLE 10: APPLICATION PROBLEM The population size of a bacterial culture growing under

controlled conditions is doubling each day. Predict how large the culture will be at the beginning of day 7 if it measures 10 units at the beginning of the day 1.

What do we know?

The bacterial culture should measure 640 units at the beginning of day 7.

HOMEWORK

Unit 19 HW Page # 6 – 13