Aim: Arithmetic Sequence Course: Math Literacy Do Now: Aim: What is an arithmetic sequence? Write the first five terms of the sequence where rule for

Aim: Arithmetic Sequence Course: Math Literacy

Do Now:

Aim: What is an arithmetic sequence?

Write the first five terms of the sequence where rule for the nth term is represented by n + 2

positive integers

terms of sequence n + 26543 7

recursive definition? 1 1n na a


Definition of Arithmetic Sequence

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

a1, a2, a3, a4, . . . . . an, . . .

is arithmetic if there is a number d such that

a2 – a1 = d, a3 – a2 = d, a4 – a3 = d, etc.

The number d is the common difference on the arithmetic sequence. Each term after the first is the sum of the preceding term and a constant, c.

7, 11, 15, 19, . . . . 4 4 4 4 = d

2, -3, -8, -13, . . . . -5 -5 -5 -5 = d

4n + 3, . . .

7 – 5n, . . .

finite

infinite

1st term nth term


The nth Term of an Arithmetic Sequence

The nth term of an arithmetic sequence has the form

an = dn + c

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

c = a1 – d

Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2.a1 = 2

an = dn + c an = 3n – 1

2 = 3(1) + can = dn + c c = -1

2, 5, 8, 11, 14, . . . , 3n – 1, . . .

An alternative form of the nth term is an = a1 + (n – 1)d

7, 11, 15, 19, . . . . 4 4 4 4 = d

4n + 3, . . .


Recursive Formula for Arithmetic Sequence

1n na a d

Recursive definition of a sequence – if 1 or more of the first terms are given – all other terms are defined by using the previous term(s) with a formula.

positive integers

terms of sequence n + 26543 7

recursive definition? 1 1n na a


Model Problem

Find the twelfth term of the arithmetic sequence 3, 8, 13, 18, . . . .

d = 5 an = dn + c

c = a1 – d

an = a1 + (n – 1)d

a12 = ?

a12 = 3 + (12 – 1)5

a12 = 3 + (11)5 = 58

Rule? an = dn + c c = a1 – d

c = 3 – 5 = -2an = 5n – 2

18 = 5(4) – 2 check:


Model Problem

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

a13 = 9d + a4

65 = 9d + 20

a4 = 5(4) + c 20 = 5(4) + c

an = 5n + 0

1 2 3 4 5 6

5, 10, 15, 20, 25, 30, . . .

an = dn + c

5 = d

0 = c

an = dn + c


Model Problem

According to the National Education Association, teachers in the United States earned an average of $30,532 in 1990. This amount has increased by approximately $1472 per year.a.Write a formula for the nth term of the arithmetic sequence that describes teachers’ average earnings n years after 1989.

b.How much will US teachers earn by the year 2010?

a1 = 30,532, d = 1472

an = a1 + (n – 1)d

an = 30,532 + (n – 1)1472n = 21

a21 = 30,532 + (20)1472 = 59,972


Model Problem

According to the US Census Bureau, new one-family houses sold for an average of $159,000 in 1995. This average sales price has increased by approximately $9700 per year.a.Write a formula for the nth term of the arithmetic sequence that describes the average cost of new one-family houses n years after 1994.

b.How much will new one-family houses cost, on average, by the year 2010?

a1 = 159,000, d = 9700

an = a1 + (n – 1)d

an = 159000 + (n – 1)9700

n = 16

a21 = 159000 + (15)9700 = 304,500


Arithmetic Means

The terms between any two nonconsecutive terms of an arithmetic sequence are called arithmetic means. In the sequence below, 38 and 49 are the arithmetic means between 27 and 60.

5, 16, 27, 38, 49, 60

arithmetic means

between 27 & 60

simple example: insert one arithmetic mean between 16 and 20 an = a1 + (n – 1)d

20 = 16 + (2)d

d = 2

an = a3 = 20a1 = 16(n – 1) = 3 – 1 = 2

16, 18, 20


Model Problem

Write an arithmetic sequence that has five arithmetic means between 4.9 and 2.5.

an = a1 + (n – 1)d

2.5 = 4.9 + (6)d

d = -0.4

an = a7 = 2.5

a1 = 4.9

n = 7

4.9, ___, ___, ___, ___, ___, 2.5

a2 = 4.9 + (-0.4) = 4.5

4.5

a3 = 4.5 + (-0.4) = 4.1a4 = 4.1 + (-0.4) = 3.7

a5 = 3.7 + (-0.4) = 3.3

a6 = 3.3 + (-0.4) = 2.9

4.1 3.7 3.3 2.9

Documents

Aim: Arithmetic Sequence Course: Math Literacy Do Now: Aim: What is an arithmetic sequence? Write the first five terms of the sequence where rule for