lesson 8 - weeblygvmath.weebly.com/uploads/5/6/5/3/56539921/chapter...lesson 8.3 • arithmetic and...

Counting techniques and finite differences are two methods that can beused to find closed-form solutions for recurrence relations. However, thereare many kinds of recurrence relations, and no method is capable offinding a closed-form solution for all of them.

Two of the most common types of recurrence relations are those inwhich a term is generated by either adding a constant to the previousterm or multiplying the previous term by a constant. The first type iscalled arithmetic, and the constant is called the common difference. Thesecond type is called geometric, and the constant is called the commonratio. This lesson considers arithmetic and geometric recurrence relationsand a few of their applications.

A surprising fact about these two types of recurrence relations is thata geometric recurrence relation with a common ratio larger than 1 and apositive first term will eventually grow to a larger value than anarithmetic recurrence relation, even if the latter’s first term and commondifference are relatively large. For example, consider a job that employsyou for 30 days and in which you have a choice of two methods ofpayment. Method 1 pays $5,000 the first day and raises the daily pay by$10,000 each day after that. Method 2 pays only $0.01 the first day butdoubles the amount you are paid each successive day. The questions are,of course: Which salary is better? How much better?

Lesson 8.3

Arithmetic and Geometric Recursion

DM08_Final.qxp:DM08.qxp 5/10/14 9:03 AM Page 464

465Lesson 8.3 • Arithmetic and Geometric Recursion

The first method of payment is arithmetic, and the commondifference is $10,000. The second method is geometric, and the commonratio is 2. If Pn is the payment on the nth day, the arithmetic recurrencerelation is Pn = Pn–1 + 10,000; the geometric recurrence relation is Pn = 2Pn–1.

To determine the amount by which one salary is better than theother, you must find the total amount each pays over the 30-day period.If Tn represents the total, then a recurrence relation that describes thetotal is Tn = Tn–1 + Pn.

Of course, all these questions can be answered by using a computeror calculator to generate a comparative table for the entire 30-day period.However, there are formulas that can find all the relevant informationdirectly. Since these formulas have many modeling applications, thislesson next discusses how the formulas can be developed.

Formulas for Arithmetic and Geometric TermsIn method 1, the pay on the first day is $5,000, the pay on the second dayis $5,000 + $10,000, and the pay on the third day is $5,000 + $10,000 +$10,000 = $5,000 + 2($10,000). Therefore, the pay on the nth day is$5,000 plus n – 1 raises of $10,000 each. Thus, the formula for the pay onthe nth day is $5,000 + (n – 1)$10,000.

In general, the nth term of an arithmetic recurrencerelation is found by adding n – 1 common differencesto the first term: tn = t1 + (n – 1)d.

In method 2, the pay on the first day is $0.01, the pay on the secondday is $0.01(2), and the pay on the third day is $0.01(2)(2) = $0.01(22).Therefore, the pay on the nth day is $0.01 doubled n – 1 times. Thus, theformula for the pay on the nth day is $0.01(2n–1).

In general, the nth term of a geometric recurrencerelation is found by multiplying the first term by thecommon ratio n – 1 times: tn = t1(r

n–1).

With these formulas, the comparison of wages on the 30th day is asimple matter:

Method 1: t30 = $5,000 + (30 – 1)$10,000 = $295,000.

Method 2: t30 = $0.01(230–1) = $5,368,709.12.


466 Chapter 8 • Recursion

The trend lines in Figure 8.2 show the wages for each method overthe 30-day period.

Figure 8.2. Daily wages by two methods.

Sums of Arithmetic and Geometric TermsA complete comparison of the two methods requires the total of the 30daily wages for each method. To examine the total pay for method 1,consider the general arithmetic recurrence relation with first term t1 andcommon difference d.

DifferencesTerm Term Sum of First n Terms First SecondNumber

1 t1 t1___ ___

2 t1 + d t1 + (t1 + d) = 2t1 + d t1 + d___

3 t1 + 2d (t1 + 2d) + (2t1 + d) = 3t1 + 3d t1 + 2d d4 t1 + 3d (t1 + 3d) + (3t1 + 3d) = (4t1 + 6d) t1 + 3d d

The constant second differences indicate that the closed-formsolution is a second-degree polynomial; tn = an

2 + bn + c. The relatedsystem of equations created from the first three pairs in the table is:

When n = 1, t1 = a + b + c.

When n = 2, 2t1 + d = 4a + 2b + c.

When n = 3, 3t1 + 3d = 9a + 3b + c.

5,000,000

3,000,000

2,000,000

1,000,000

0

4,000,000

Dai

ly W

age

($)

Day0 5 10 15 20 25 30

Method 1

Method 2



Matrices can be used to solve the system:

In general, the sum of the first n terms of anarithmetic recurrence relation is 0.5dn2 + (t1 – 0.5d)n.

The sums in method 2 do not generate constant differences in a finitenumber of steps, and, therefore, a closed-form solution cannot be foundby the finite difference method. It can, however, be found by other means.

Consider the general geometric recurrence relation with first term t1and common ratio r. The sum of the first n terms, Sn, is

Sn = t1 + t1r + t1r2 + . . . + t1r

n–1.

Multiply this equation by r:

rSn = r(t1 + t1r + t1r2 + . . . + t1r

n–1).

Distribute r on the right side of the equation, and subtract thisequation from the original equation:

Sn = t1 + t1r + t1r2 + . . . + t1r

n–1

rSn = t1r + t1r2 + t1r

3 + . . . + t1rn

Sn – rSn = t1 – t1rn.

Now factor both sides and divide by (1 – r):

Sn(1 – r) = t1(1 – rn),

orSn = .

t rr

n(1 )1

1 −−

t

t d

t d

1 1 14 2 19 3 1

2

3 3

11

2

1

++

=

−

t

d

1 1 14 2 19 3 1

1 02 13 3

1

1

=

−

t

d

dt d

0 0.51 0.50 0

0.50.5

0

11−

= −



In general, the sum Sn of the first n terms of a

geometric recurrence relation is .

With the arithmetic and geometric sumformulas, comparison of the total wages is asimple matter:

Method 1: S30 = 0.5 × 10,000 × 302 +

(5,000 – 0.5 × 10,000) × 30 = $4,500,000.

Method 2: S30 = 0.01(1 – 230)/(1 – 2) =

$10,737,418.23.

Including Sums in SpreadsheetsIf your spreadsheet has the term numbers andthe terms of a recurrence relation in columnsA and B, respectively, adding a column for thesum requires entering a single formula, thencopying it into as many cells as you need. Forexample, if the first term of the recurrencerelation is in cell B1, the second term is in B2,and so forth, type B1 in cell C1 and theformula C1 + B2 in cell C2. Copy the formulainto cells C3, C4, and so on.

Including Sums in ProgramsLesson 8.1 gives an algorithm for generating terms of the recurrencerelation Hn = Hn–1 + (n – 1) and implementations for the BASIC languageand a calculator language. To include sums in the algorithm, introduce avariable (S) for the sum, give it an initial value equal to the first term, andhave it accumulate values of the terms as the terms are generated. Beloware Lesson 8.1’s implementations with instructions to accumulate sums.

10 N = 1:H = 0:S = H 1 → N:0 → H:H → S20 PRINT N, H, S Lbl A30 N = N + 1 Disp N, H, S40 H = H + N – 1 N + 1 → N:H + N – 1 → H:S + H → S50 S = S + H Goto A60 GO TO 20

t r

r

n1 1

1

( )−

−

Lighthammer ContinuesGeometric Growth

Clarinet NewsApril 9, 2003

Lighthammer, the leading supplier ofmanufacturing intelligence solutions,announced first quarter 2003 resultsending March 31, 2003 showing anincrease of 385% over first quarter2002. Russ Fadel, Lighthammer CEOcommented, “We are pleased with thetremendous year-to-year growth of ourbusiness particularly with the weakeconomy and the political uncertaintycreated by the war in Iraq. Ourgeometric growth is directly attributedto the operational performanceimprovement enabled by ourmanufacturing intelligence solution.”



Keep in mind that neither algorithm terminates unless additionalinstructions are added.

Sums on a Graphing CalculatorSome graphing calculators have functions that automatically calculate sums. The screen on the left shows the closed-form solution tn = 0.01 × 2

n–1 as it is used to calculate the 30 daily wages by method 2.The results are stored in one of the calculator’s lists. The second screenshows the sum function as it is used to find the sum of the 30 daily wages.

In this lesson’s exercises you have the opportunity to use yourknowledge of arithmetic and geometric terms and sums to model avariety of situations.

Exercises1. Consider the following sequences:

i. 2, 5, 8, 11, 14, . . .ii. 64, 32, 16, 8, 4, 2, 1, . . .iii. 10, 12, 14.4, 17.28, 20.736, . . .iv. 2, 3, 5, 8, 13, 21, . . .v. , , , , . . .

vi. 3, 4, 6, 9, 13, 18, . . .

a. Which of the sequences are geometric, which are arithmetic,and which are neither?

b. Write a recurrence relation for each sequence.

c. For those sequences that are arithmetic or geometric, write aclosed-form formula for the nth term.

310,000

31000

3100

310



2. Consider the general recurrence relation for a geometric sequence,tn = rtn–1. Find the fixed point for this recurrence relation if oneexists. (Recall the hint in Exercise 4b, page 461, of Lesson 8.2.)

3. Consider the general recurrence relation for an arithmeticsequence, tn = tn–1 + d. Find the fixed point for this recurrencerelation if one exists.

4. 900 telephone numbers charge the caller for thecall. Consider a 900 teen chat line that charges$2 for the first minute and $0.95 for eachadditional minute.

a. What recurrence relation does the statementof the problem suggest? (Make a table ifnecessary.)

b. Find a closed-form formula to describe the costof a call.

c. Assume that 5,000 teens use the line each week and talk for anaverage of 15 minutes. How much income is produced? Supposealso that the bulk of the cost to the company operating the lineis the cost of the long distance line, which averages $3 for a 15-minute call. How much profit does the company make each week?

5. George deposits $5,000 in the bank at an annual interest rate of4.8% compounded yearly.

a. Write a recurrence relation for the amount of money in theaccount at the end of n years.

b. Write a closed-form formula for the amount of money in theaccount at the end of n years.

c. How much money is in George’s account at the end of 3 years?

d. The bank decides to become competitive with other bankinginstitutions in the city by offering an interest rate of 4.8% thatcompounds monthly. What monthly interest rate does Georgereceive now?

e. Write a recurrence relation for the amount of money in theaccount at the end of n months.



f. Write a closed-form formula for the amount of money in theaccount at the end of n months.

g. Find the amount in George’s account at the end of 3 years (36 months).

h. Compare the amount of money in George’s account at the endof 3 years when the interest is compounded yearly with theamount at the end of 3 years when the interest is compoundedmonthly.

6. In Exercise 5, suppose George is given the choice of investing his$5,000 at 4.8% annual interest compounded monthly or at 5.0%annual interest compounded yearly. Compare the two methods ofinvestment at the end of 1 year. At the end of 2 years. At the end of 3 years.

7. Find the fifteenth term and the sum of the first 15 terms for thesequences in parts a to c.

a. 4, 9, 14, 19, . . .

b. 45.75, 47, 48.25, 49.5, . . .

c. 3650, 3623, 3596, 3569, 3542, . . .

d. From your work in other mathematics courses, you may befamiliar with the use of the Greek letter Σ (sigma) to indicatesums. For example, if you are summing the first five terms of thesequence whose terms are generated by 2n – 3, the sum can be

indicated in this way: (2n – 3). The term numbers of the first

and last terms that are included in the sum are written at thebottom and at the top of sigma, respectively, and the formula forthe nth term is written in parentheses to the right. Use this typeof summation notation to represent each of the sums you foundin parts a to c.

8. The number of deer on Fawn Island is currentlyestimated at 500 and is increasing at a rate of8% per year. At the present time, the island cansupport 4,000 deer, but acid rain is destroyingthe vegetation on the island and the number ofdeer that can be fed is decreasing by 100 peryear. In how many years will there not beenough food for the deer on Fawn Island?Explain how you got your answer.

1

5∑



9. The cost of n shirts selling for $14.95 each is given by Cn = 14.95n.

a. What is the equivalent recurrence relation?

b. Is the recurrence relation arithmetic, geometric, or neither? If itis arithmetic or geometric, what are the first term and thecommon difference or ratio?

10. At 5.5% compounded yearly, what amount must Bill’s parentsinvest for Bill at age 10 if they want Bill to be a millionaire when hereaches age 50?

11. To double your investment at the end of 11 years, what annualinterest rate do you need to receive?

12. The following table gives data relating temperature in degreesFahrenheit to the number of times a cricket chirps in one minute.Although the data are not quite a perfect arithmetic sequence, theyare close.

Temperature (°F) 50 52 55 58 60 64 68

Chirps per minute 40 48 60 73 80 98 114

a. Assuming that the data form an arithmetic sequence, with thetemperature as the term number and the chirps per minute asthe term value, what is the common difference per degree rise intemperature?

b. What is the temperature when a cricket is chirping 110 times perminute?

c. At what temperature does a cricket stop chirping?

By permission of John L. Hart FLP, and Creators Syndicate, Inc.



13. A ball is dropped from a height of 8 feet andrebounds on each bounce to 75% of itsheight on the previous bounce.

a. How high does the ball reach after thesixth bounce?

b. What is the total distance that the ballhas traveled just before the seventhbounce?

14. The tenth term of an arithmetic sequence is4, and the twenty-fifth term is 20. Find thefirst term and the common difference for thisarithmetic sequence.

15. The average cost of a year of collegeeducation in a public university is about$16,500. The cost is increasing an average of 5% per year.

a. If the current ratecontinues, what willbe the cost of a yearof college educationin 30 years?

b. Ten years from nowyou start saving forthe college educationof a child. The bestinterest rate you canget is 6%compoundedannually. Whatamount would youhave to put in thebank in order to payfor a year of college20 years later?

A rubber ball rebounds to a fraction of theheight of the previous bounce, as shown in thistime-lapse photograph.

The Rising Cost of College

Evansville Courier & PressNovember 26, 2012

Many parents dream ofthe day their child stridesto the stage to receive acollege diploma. On theother hand, many parentsdread the day that collegetuition bills arrive. Despitesteep tuition costs, acollege education may bea sound investment that'swithin the reach of manyfamilies. All it takes is adegree of planning.

Tuition costs have beenincreasing at an averageof 5 percent annually in

recent years. As a result,four years at a publiccollege or university fortoday's newborns couldtip the scales at more than$145,000. Given thesestatistics, it is easy tobelieve that in 2009,contemporary collegestudents graduated withan average $24,000 indebt 1. Unheard of inprevious generations, thisdebt burden is nowcommonplace.



16. Over time, the number of bacteria in a culture grows geometrically.There are 600 bacteria at t = 0 and 900 bacteria at t = 3 hours.

a. What is the approximate common ratio for this sequence?

b. If the growth rate continues, approximately how many bacteriaare there after 10 hours?

c. At what time will there first be 50,000 bacteria in the culture?

17. A 50-meter-long pool is constructed with the shallow end 0.85meter deep. For each meter of length, starting at the shallow end,the pool deepens by 0.06 meter.

a. How deep is the deepest part of the pool? Explain.

b. A rope is to be placed across the pool where the pool depth is 1.6meters in order to mark the end of the shallow section. How farfrom the shallow end of the pool should the rope be placed?

18. A company has sales of $6 million one year and $8 million the nextyear. The company expects to experience “geometric growth” at asimilar rate in succeeding years.

a. Write a recurrence relation that describes the company’s growthand find a solution to the recurrence relation.

Pool

DeepShallow Top View

Side View0.85 m

?

50 m



b. Predict the company’s sales 5 years from now and 10 years from now.

c. Do you think the predictions are realistic? Explain.

19. Read the news article on page 468.

a. If a company has a 385% revenue increase in one year, what isthe geometric growth rate?

b. If a company has $200 million annual revenue and then has twoyears of growth at the rate you found in part a, what is thecurrent revenue?

Computer/Calculator Explorations20. The fractal curve construction described in

Exercise 12 of Lesson 8.1 on page 453 can beapplied to the sides of an equilateral triangle toproduce a fractal that is often called asnowflake curve. At right is one stage of such aconstruction. Adapt the procedure youdeveloped in either Exercise 13 or Exercise 14 inLesson 8.1 to produce several snowflake curves.

Projects21. Some geometric recurrence relations can be used to generate

infinite sequences with finite properties. For example, therecurrence relation tn = tn–1 with t1 = 1 generates the sequence

1, , , , . . . . The terms of this sequence approach 0, and the

sequence of sums: 1, 1 + , 1 + + , . . . approaches 2. Research

and report on conditions for which the sequence of terms and thesequence of sums of a geometric recurrence relation approach afinite value. Apply the results of your investigation to the perimeterand area of a snowflake curve based on an equilateral trianglewith sides of length 1 (see Exercise 20).

22. In this lesson, the closed-form formulas for terms and sums ofarithmetic and geometric recurrence relations are proved byalgebraic symbol manipulation. Prepare a report showing howthese formulas can be proved by mathematical induction.

12

18

14

12

14

12

12


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