Section 1.3 Arithmetic Sequences

An arithmetic sequence is a sequence in which

the difference between each term and the

preceding term is always constant.

Which of the following sequences is arithmetic?

a. {14, 10, 6, 2, -2, -6, -10, . . . }

b. {3, 5, 8, 12, 17, . . . }

a. yes, the difference between each term is -4

b. no, the difference between the first two terms is 2

and the difference between the 2nd and 3rd term is 3.

Recursive Form of an Arithmetic

Sequence

Un = Un-1 + d

for some constant d and all n > 2

The number d is called the common difference of the

arithmetic sequence.

Graph of an Arithmetic Sequence

• If {Un} is an arithmetic sequence with U1 = 3

and U2 = 4.5 as its first two terms,

• a. Find the common difference.

• b. Write the sequence as a recursive

function.

• c. Give the first six terms of the sequence.

• d. Graph the sequence.

•

a. Find the common difference.

• U – U = 4.5 – 3 = 1.5

2 1

• The common difference is 1.5

• b. Write the sequence as a recursive function.

•U 1 = 3, Un = Un-1 + 1.5, for n > 2

First Method for Always one

Term finding nth term greater than

by using the subscript of first

preceding term. term.

Explicit Form of an Arithmetic Sequence

• In an arithmetic sequence {Un} with common

difference d, Un = U1 + (n – 1)d for every n > 1

Find the nth term of an arithmetic sequence with first term -5 and

common difference of 3. Sketch a graph of the sequence.

Un = U1 + (n – 1)d

= -5 + (n – 1)3

= -5 + 3n – 3

= 3n – 8

Find the nth term of an arithmetic sequence with first term -5 and

common difference of 3. Sketch a graph of the sequence.

Finding a Term of an Arithmetic

Sequence

What is the 45th term of the arithmetic sequence whose first

three terms are 5, 9, and 13?

First find d; d = 9 – 5 = 4

Second find explicit form: Un = 5 + (n – 1)4

Un = 4n + 1

Then find 45th term: U45 = 4(45) + 1

U45 = 181

Finding Explicit and Recursive

Formulas

If {Un} is an arithmetic sequence with U6 = 57 and

U10 = 93, find U1, a recursive formula, and an explicit

formula for Un.

To find d when given to non-consecutive terms use the formula:

d = Um –Un

m n

d = 93 — 57

10 6

=9

Finding U1

Select either of the given terms and substitute

into Explicit formula. Un = U1 + (n – 1)d

U6 = 57 U10 = 93

57 = U1 + (6 – 1)9 93 = U1 + (10 – 1)9

U1 = 12 U1 = 12

FORMULAS

Explicit Form Recursive Form

Un = U1 + (n – 1)d Un = Un-1 + d

Un = 12 + (n – 1)9 Un = Un-1 + 9, for n > 2

Un = 9n + 3, for n > 1