Pre Calculus 11 Section 1.1 Arithmetic Sequences

PRE CALCULUS 11SECTION 1.1 ARITHMETICSEQUENCES

I) ARITHMETIC SEQUENCES:

A sequence is an ordered list of objects or numbers

An “arithmetic sequence” is a list of numberswhere each term increases or decreases by acommon difference (d) [same value]

Ex: Find the Common Difference:

i) 3, 6, 9, 12,….

ii) – 5, – 11, – 17, ….

iii) 19, 11, 4, -2,….

The difference between the terms are not consistent

Patterns in Arithmetic sequences are seen in everythingie: wage increase, costs vs time, cost vs quantity

Jason works at the PNE selling water. For every bottlewater he sells, he earns 35cents as commission.Suppose he earns $64 a day plus commission, how muchwill he earn if he sold 200 bottles? How many bottleswill he need to sell to earn $300?

For every 3 minutes in a taxi ride, the cost increases by$1.35. If the initial cost is $5, how much will it cost for9min? 12min? And 15min? Or if you have only $25, howlong can you afford?

In this section, you will learn to find the value of eachterm in an arithmetic sequence, how many terms thereare, common difference, and how to apply it in realapplications

Suppose Jason works at the PNE and he earns commissionon his sales. If he sold 200 bottles, he earns $134. If hesold 300 bottles he earns $169. How much commissiondoes he earn from each bottle?

After sitting in a taxi for 9 minutes, Sally realizes the costof the ride was $9.05. Three minutes later the cost was$10.40. How much will the ride be in 30minutes?

WORKING WITH ARITHMETIC SEQUENCES

Ex: In each of the sequences below, indicate whether if it isan arithmetic sequence. If yes, find the commondifference and the value of the 10th term

i) 13, 21, 29,....

ii) -6, -10, -14,....

iii) 4, 10, 17, …..

Yes it is a A.S. because each term isincreasing by 8 (common difference)

Yes it is a A.S. because each term isdecreasing by 4 (common difference: – 4 )

No it is not a A.S. because difference isnot consistent!

HOW DOES AN ARITHMETIC SEQUENCE WORK?

The first term of an A.S. is called “a” or “t1”

Each term after the first adds another common difference “d”

The value of each term is denoted “tn”, where “n” is the order ofthe term

The number of common differences “d” in each term is one lessthan it “n” value

Ex: Given the sequence: 6, 11, 16… find the 32nd term.

Write down the formula:

Identify parts you know:

Substitute the values into formula and calculate:

tn = a + (n – 1)d

= 6 + (32 – 1)(5)

= 6 + (31)(5)

= 6 + 155

= 161

The 32nd term is 161

PRACTICE: SOLVE EACH OF THE PROBLEMS BELOW

Find the 30th term in the sequence:

Find the 200th term in the sequence:

Ex: An arithmetic sequence is 53, 49, 45, . . . One term inthis sequence is -107. Which term is it?

Write down the formula: tn = a + (n – 1)d

Identify parts you know:

n = ?, a = 53, d = 49 – 53 = -4, tn = -107

Sub parts into formula and calculate:

tn = a + (n – 1)d

– 107 = 53 + (n – 1)(– 4)

– 107 = 53 – 4n + 4

– 107 = 57 – 4n

– 164 = – 4n

41 = n

Notice the –ve sign. Why? Because of the

Distributive property

PRACTICE: SOLVE EACH OF THE PROBLEMS BELOW

What term is -523 in the arithmetic sequence?

Find the 3 missing terms in the arithmetic sequence:

–523 is the65th term!

Therefore, the numbers will be:

Ex: In an arithmetic sequence, the 5th term is 7 and the12th term is -49

a.List the sequence and show the first 5 terms.

b.Write the general term for the sequence.

c.How many terms are greater than -100?

List the sequence first

_, ___, __, __, 7, ___, ___, ___, ___, ___, ___, -49

Find the common difference

To find previous terms, justsubtract the common difference.

Work backwards!!

b.Write down the formula: tn = a + (n – 1)d

List parts you know:a = 39, d = -8

Sub parts into formula:

tn = 39 + (n – 1)(-8)

= 39 – 8n + 8Dist. Prop causes thesign change

= -8n + 47This is the generalterm

Do a check by finding t5

t5 = -8n + 47

= -8(5) + 47

= -40 + 47

= 7

c.To find terms greater than -100, you needsome kind of inequality.

Set up the inequality, then solve for “n”

-8n + 47 > -100

-8n > -100 – 47

-8n > -147

n < -147

-8

n < 18.375

Notice the inequality “switched” when

divided by a –ve number

Since 18.375 will give you exactly -100, only 18 termsis allowed because the # of terms must be smallerthan 18.375 (besides, you can’t have “.375” of a term)

HW:

Assignment 1.1 Arithmetic sequences

Optional assignment p16 #1-6 odd letters, 8, 10,11, 13, 17, 18