Arithmetic and Arithmetic and

Geometric SequencesGeometric Sequences

by Pam Tobeby Pam Tobe Beth BosBeth Bos Mary Lou SheltonMary Lou Shelton

• Suppose you have $40 in a piggy Suppose you have $40 in a piggy bank that you are saving to spend on bank that you are saving to spend on a special project. You take on a a special project. You take on a part-time job that pays $13 per day. part-time job that pays $13 per day. Each day you put the cash into the Each day you put the cash into the piggy bank. The number of dollars in piggy bank. The number of dollars in the bank is a function of the number the bank is a function of the number of days you have worked.of days you have worked.

Days Days

nn

DollarsDollars

ttnn

11 5353

22 6666

33 7979

44 9292

55 105105

But it is a But it is a discrete discrete functionfunction rather than a rather than a continuous functioncontinuous function. . After 3 ½ days you still After 3 ½ days you still have the same $79 as have the same $79 as you did after 3 days. A you did after 3 days. A function like this, whose function like this, whose domain is the set of domain is the set of positive integers, is positive integers, is called a called a sequencesequence..

Objectives:Objectives:• Represent sequences Represent sequences

explicitly and recursivelyexplicitly and recursivelyGiven information about a sequenceGiven information about a sequence

• Find a term when given its term Find a term when given its term numbernumber

• Find the term number of a given Find the term number of a given term.term.

Let’s investigate the sequence of Let’s investigate the sequence of dollars 53, 66, 79, 92, 105… in the dollars 53, 66, 79, 92, 105… in the previous problem by:previous problem by:

a.a. Sketching the graph of the first few Sketching the graph of the first few terms of the sequence.terms of the sequence.

b.b. Finding tFinding t100n100n the 100 the 100thth term of the term of the sequencesequence

c.c. Writing an equation for tWriting an equation for tnn the nth the nth term of the sequence, in terms of n.term of the sequence, in terms of n.

a) a) The graph shows The graph shows discrete point. You discrete point. You may connect the may connect the points with a dashed points with a dashed line to show the line to show the pattern, but don’t pattern, but don’t make it a solid line make it a solid line because sequences because sequences are defined on the are defined on the set of natural set of natural numbers.numbers.

b. To get the fourth term, you add the b. To get the fourth term, you add the common difference of 13 three times common difference of 13 three times to 53. So to get the 100to 53. So to get the 100thth term, you term, you add the common difference 99 times add the common difference 99 times to 53.to 53.

tt100n100n = 53 + = 53 + 9999(13) = 1340(13) = 1340

c. tc. tnn = 53 +13( = 53 +13(n-1n-1))

• The sequence in Example 1 is called The sequence in Example 1 is called an an arithmetic sequencearithmetic sequence. You get . You get each term by adding the same each term by adding the same constant to the preceding term. You constant to the preceding term. You can also say that the difference of can also say that the difference of consecutive terms is a constant. The consecutive terms is a constant. The constant is called the constant is called the common common differencedifference..

• The pattern “add 13 to the previous term The pattern “add 13 to the previous term to get the next term” in Example 1 is to get the next term” in Example 1 is called a recursive pattern for the called a recursive pattern for the sequence. You can write an algebraic sequence. You can write an algebraic recursion formularecursion formula

ttn =n = t tn-1n-1 + 13 + 13

Sequence mode:Sequence mode:

nMin = 1 beginning value of nnMin = 1 beginning value of n

u(n) = u(n-1)+13 recursion formulau(n) = u(n-1)+13 recursion formula

u(uMin) = {53} enter first termu(uMin) = {53} enter first term

Press GraphPress Graph

The pattern tn = 53 + 13(n-1) is called an explicit formula for the sequence. It “explains” how to calculate any desired term without finding the terms before it.

Arithmetic Sequence:Arithmetic Sequence:A sequence in which consecutive terms differ by a A sequence in which consecutive terms differ by a fixed amount is an arithmetic sequence, or arithmetic fixed amount is an arithmetic sequence, or arithmetic progressionprogression..

Definition: Definition: Arithmetic SequenceArithmetic Sequence

A sequence (aA sequence (ann) is an ) is an arithmetic arithmetic

sequencesequence (or arithmetic progression) if it (or arithmetic progression) if it can be written in the form:can be written in the form:

aann = a = an-1n-1 + + dd n n> > 22

For some constant For some constant dd. The number . The number dd is is the the common differencecommon difference..

Pair/Share, Try ItPair/Share, Try It

Determine whether the sequence Determine whether the sequence could be arithmetic. If so, find the could be arithmetic. If so, find the common difference.common difference.

a)a) -6, -3.5, -1, 1.5, 4,……-6, -3.5, -1, 1.5, 4,……

b)b) 48, 24, 12, 6, 3,…..48, 24, 12, 6, 3,…..

c)c) In3, In 5, In12, In24In3, In 5, In12, In24

If (aIf (ann ) is an arithmetic sequence with ) is an arithmetic sequence with

common difference d, thencommon difference d, then

aa2 2 = a= a1 1 + d+ d

aa3 3 = a= a22 + d = a + d = a1 1 + 2d+ 2d

aa44 = a = a3 3 + d = a+ d = a1 1 + 3d+ 3d

nth Term of an Arithmetic Sequencenth Term of an Arithmetic Sequence

The nth term of an arithmetic sequence can be The nth term of an arithmetic sequence can be written in the form: written in the form: aann = a = a1 1 + (n – 1)d+ (n – 1)d

Where Where aa11 is the first term and is the first term and dd is the common is the common differencedifference

Pair/Share, Try ItPair/Share, Try It

The third and eighth terms of an The third and eighth terms of an arithmetic sequence are 13 and 3, arithmetic sequence are 13 and 3, respectively. Find the first term, the respectively. Find the first term, the common difference, and an explicit common difference, and an explicit rule for the nth term.rule for the nth term.

Geometric SequenceGeometric Sequence

In an In an arithmetic sequencearithmetic sequence, terms are , terms are found by adding a constant to the found by adding a constant to the preceding term. A sequence in preceding term. A sequence in which terms are found by multiplying which terms are found by multiplying the preceding term by a (nonzero) the preceding term by a (nonzero) constant is a constant is a geometric sequence or geometric sequence or geometric progressiongeometric progression..

Definition: Definition: Geometric SequenceGeometric SequenceA sequence (an) is a A sequence (an) is a geometric sequencegeometric sequence (or (or geometric progressiongeometric progression) if it can be written in the ) if it can be written in the formform

aan n = a= an-1n-1 * r, n * r, n >> 2 2

Where r ≠ 0 is the Where r ≠ 0 is the common ratio.common ratio.

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Determine whether the sequence could Determine whether the sequence could be geometric. If so, find the common be geometric. If so, find the common ratio.ratio.

-3 -1 1 3 5

2 2 2 2a) 2, , , , .........3 9 15 21

b) 3, 6, 12, 24, 48 .......

c) 10 , 10 , 10 , 10 , 10 .......

nth Term of a Geometric Sequencenth Term of a Geometric Sequence

The nth term of a geometric sequence can The nth term of a geometric sequence can be written in the form.be written in the form.

aann = a = a11 * r * rn-1n-1

Pair /Share, Try ItPair /Share, Try ItThe third and eighth terms of a geometric The third and eighth terms of a geometric sequence are 20 and -640, respectively. sequence are 20 and -640, respectively. Find the first term, common ratio, and an Find the first term, common ratio, and an explicit rule for the nth term.explicit rule for the nth term.

ApplicationApplication

• The population of Bridgetown is growing at The population of Bridgetown is growing at the rate of 2.5% per year. The present the rate of 2.5% per year. The present population is 50,000. Find a sequence that population is 50,000. Find a sequence that represents Bridgetown’s population each represents Bridgetown’s population each year. Represent the nth term of the year. Represent the nth term of the sequence both explicitly and recursively. sequence both explicitly and recursively. Evaluate seven terms of the sequence.Evaluate seven terms of the sequence.

StateState PopulationPopulation

(1993)(1993)

Growth rateGrowth rate

New YorkNew York 18,197,00018,197,000 0.5%0.5%

TexasTexas 18,031,00018,031,000 2.0%2.0%

Assume that the population of New York and Assume that the population of New York and Texas continued to grow at the annual Texas continued to grow at the annual rate as shown:rate as shown:

a)a) In what year will the population of Texas In what year will the population of Texas surpass that of New York?surpass that of New York?

b)b) In what year will the population of Texas In what year will the population of Texas surpass that of New York by 1 million?surpass that of New York by 1 million?