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1 8. Airthmetic and Geometric Sequences ARITHMETIC AND GEOMETRIC SEQUENCES Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I keep returning to the same problem. Why does the mathematics we have discovered in the past so often turn out to describe the workings of the Universe? John Barrow Two kinds of regular sequences occur so often that they have specific names, arithmetic and geometric sequences. We treat them together because some obvi ous parallels between these kinds of sequences lead to similar formulas. This also makes it easier to learn and work with the formulas. The greatest value in this association is understanding how the ideas are related and how to derive the formulas from fundamental concepts. Anyone learning the formulas this way can recover them whenever needed. Both arithmetic and geometric sequences begin with an arbitrary first term, and the sequences are generated by regularly adding the same number (the common difference in an arithmetic sequence) or multiplying by the same number (the common ratio in a geometric sequence). Definitions emphasize the parallel features, which examples will clarify. I remember that when I was about twelve I learned from [my uncle] that by the distributive law times equals. I thought that was great. Peter Lax Definition: arithmetic and geometric sequences Arithmetic Sequence a a and a n a n d for n The sequence a n is an arithmetic sequence with first term a and common difference d. Geometric Sequence a a and a n r a n for n The sequence a n is a geometric sequence with first term a and common ratio r. The definitions imply convenient formulas for the nth term of both kinds of sequences. For an arithmetic sequence we get the nth term by adding d to the first term n times; for a geometric sequence, we multiply the first term by r, n times. Formulas for the nth terms of arithmetic and geometric sequences For an arithmetic sequence, a formula for the nth term of the sequence is a n a n d. () For a geometric sequence, a formula for the nth term of the sequence is a n a r n. () The definitions allow us to recognize both arithmetic and geometric sequences. In an arithmetic sequence the difference between successive terms, a n a n,is always the same, the constant d; in a geometric sequence the ratio of successive terms, a n, is always the same. a n
2 45 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers EXAMPLE Arithmetic or geometric? Thefirstthreetermsofasequence are given. Determine if the sequence could be arithmetic or geometric. If it is an arithmetic sequence, find d; for a geometric sequence, find r. Strategy: Calculate the differences and /or ratios of successive terms. (a),4,8,... (b) ln,ln4,ln8,... (c),, 4,... (a) a a 4, and a a Since the differences are not the same, the sequence cannot be arithmetic. Checking ratios, a 4 a, and a 8, so the sequence could be geometric, with a common ratio a 4 r. Without a formula for the general term, we cannot say anything more about the sequence. (b) a a ln 4 ln ln 4 ln, and a a ln 8 ln 4 ln 8 4 ln, so the sequence could be arithmetic, with ln as the common difference. As in part (a), we cannot say more because no general term is given. (c) a a,anda 6 a 4. The differences are a not the same, so the sequence is not arithmetic. a, and a 4 a, so the sequence is not geometric. Note that the sequence in part 4 (a) could be geometric and the sequence in part (b) could be arithmetic, but in part (c) you can conclude unequivocally that the sequence cannot be either arithmetic or geometric. EXAMPLE Arithmetic or geometric? Determine whether the sequence is arithmetic, geometric, or neither. (a).6n (b) n (c) a n ln n (a) a a , and a a From the first three terms, this could be an arithmetic sequence with d.6. Check the difference a n a n. a n a n.6 n.6n.6. The sequence is arithmetic, with d.6. (b) a a 4, and a a 8 4 4, so the sequence is not arithmetic. Using the formula for the general term, a n a n n n. The sequence n is geometric, with as the common ratio.
3 8. Airthmetic and Geometric Sequences 45 (c) a n a n ln n ln n ln n. The difference depends on n, so n the sequence is not arithmetic. Checking ratios, a n ln n,sotheratio a n ln n also changes with n. The sequence is neither arithmetic nor geometric. EXAMPLE Arithmetic sequences Show that the sequence is arithmetic; find the common difference and the twentieth term. (a) a n n (b) 50,45,40,...,55 5n,... (a) The first few terms of a n are,,5,7,...,fromwhich it is apparent that each term is more than the preceding term; this is an arithmetic sequence with first term and common difference a andd. Check to see that a n a n. To find a 0, use either the defining formula for the sequence or Equation () for the nth term: a or a 0 a 9d 9 9. (b) If b n 55 5n, then b n b n 55 5 n 55 5n 5. This is an arithmetic sequence with a 50, d 5, and so b Given the structure of arithmetic and geometric sequences, any two terms completely determine the sequence. Using Equation () or (), two terms of the sequence give us a pair of equations from which we can find the first term and either the common difference or common ratio, as illustrated in the next example. EXAMPLE 4 Arithmetic sequences Suppose a n is an arithmetic sequence with a 8 6anda 4. Find a, d, and the three terms between a 8 and a. From Equation (), a 8 a 7d, anda a d, from which the difference is given by a a 8 4d. Use the given values for a 8 and a to get 4 6 4d, or d 5. Substitute 5 47 for d in 6 a 7d and solve for a, a.findthe three terms between a 8 and a by successively adding 5 : a 9 a 8 5 7, a 0 a 9 5, a a 0 5. Therefore, a 9 is 7, a 0 is, and a is. EXAMPLE 5 Geometric sequences Determine whether the sequence is geometric. If it is geometric, then find the common ratio and the terms a, a, and a 0. (a) n (b),, 9,..., n,...
4 454 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers Strategy: The property that identifies a geometric sequence is the common ratio: the values a, a, a 4,... a a a must all be the same. For a geometric sequence, use Equation (). (a) Thefirstfewtermsare,4,8,6,...,each of which is twice the preceding term. This is a geometric sequence with first term a, and common ratio given by r a n a n n. Using a n n n, (b) Consider the ratio a a 8 and a a n a n n n, so the sequence is geometric with a andr. Using a n n, we get a, a ar 9,anda 0 ar Partial Sums of Arithmetic Sequences There is a charming story told about Carl Freidrich Gauss, one of the greatest mathematicians of all time. Early in Gauss school career, the schoolmaster assigned the class the task of summing the first hundred positive integers, That should have occupied a good portion of the morning, but while other class members busied themselves at their slates calculating, 6, 6 4 0, and so on, Gauss sat quietly for a few moments, wrote a single number on his slate, and presented it to the teacher. Young Gauss observed that and 00 add up to 0, as do the pair and 99, and 98, and so on up to 50 and 5. There are fifty such pairs, each with a sum of 0, for a total of , the number he wrote on his slate. This approach works for the partial sum of any arithmetic sequence, and we will use the method to derive some useful formulas. However, the ideas are more valuable than memorizing formulas. If you understand the idea, you can recreate the formula when needed. To find a formula for the nth partial sum of an arithmetic sequence, that is, the sum of n consecutive terms, pair the first and last terms, the second and nexttolast, andsoon;each pair has the same sum. Infact,itiseasiertopairalltermstwice, as illustrated with Gauss sum: S S S The sum on the right has 00 terms, so S Dividing by, S For the general case, pairing the terms in S n and adding gives S n n a a n because there are n pairs, each with the same sum. Dividing by yields the desired formula.
5 8. Airthmetic and Geometric Sequences 455 Partial sums of an arithmetic sequence Suppose a n is an arithmetic sequence. The sum S n of the first n terms is given by S n n a a n () The formula is probably most easily remembered as n times the average of the first and last terms. EXAMPLE 6 Partial sums For the sequence a n n, (a) evaluate the sum S 5 5 k k and (b) find a formula for S n. Strategy: Let a Follow the strategy. n n. To find S 5 from Equation () requires a and a 5, (a) By Equation (), S 5 5 a a 5.Now,finda and a 5. which the formula for a n can provide. For (b), substitute for a a and a and n fora n in Equation () and simplify Thus, S (b) In general, S n n a a n Hence, S n n. n n n n n. EXAMPLE 7 Arithmetic sequence Thesumofthefirsteighttermsofan arithmetic sequence a n is 4; the sixth term is 0. Find a formula for a n. For a n, first find a and d. Since a 6 a 5d, a 5d 0. Express S 8 in terms of a and d, 8 a a 7d S 8 4 a 7d. Since we are given S 8 4, Equation () states that 4 a 7d 4. This gives a pair of equations to solve for a and d. a 5d 0 a 7d 6 We find d anda 0. Therefore, the nth term is a n a n d 0 n n. Partial Sums of Geometric Sequences The idea of pairing terms, which works so well for arithmetic sequences, does not help with a geometric sequence. Another idea does make the sum easy to calculate
6 456 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers though. Multiply both sides by r and subtract: S n a ar ar ar n rs n ar ar ar n ar n S n rs n a ar n Thus, S n r a r n. If r, dividing both sides by r yields a formula for S n. Partial sums of a geometric sequence Suppose a n is a geometric sequence with r. The sum of the first n terms is S n a r n r (4) In the special case where r, the geometric sequence is also an arithmetic sequence, and S n a a a a na. Strategy: Since it is given that the sequence is a geometric, find the common ratio r a andthenuse a Equations () and (4). EXAMPLE 8 Partial sum Find a n and S n for the geometric sequence,, 6 Follow the strategy. We know that a and a is 6. The common ratio is r a 6 a n ar n. From Equation (), n n. Since r, r and r n n. Applying Equation (4) gives S n a r n r Therefore, n a n n and S n n n. EXAMPLE 9 Limit of a sum (a) Find the sum of the first 5, 0, and 00 terms of the geometric sequence from Example 8. (b) Draw a graph of S n in 0, c 0,, where c is the number of pixel columns of your n calculator (see inside front cover). Trace to find the smallest integer n for which the yvalue is displayed as.
7 8. Airthmetic and Geometric Sequences 457 [0, c] by [0, ] FIGURE y (/)( / x) (a) InExample8wefoundaformulaforthenth partial sum, S n. Substituting 5, 0, and 00 for n, n S , S S The term has 0 zeros immediately following the decimal point. That 00 means that S 00 is so near that a calculator cannot display the difference except as. (b) In the window 0, c 0, we see a graph something like Figure. Because calculators display trace coordinates differently, you may see something other than ours, but somewhere between 5 and 5, you should see the yvalue displayed something like , the nearest your calculator can come to displaying. Looking Ahead to Calculus: Infinite Series As indicated above, each sequence a n is associated with a sequence of partial sums S n, where S n a a a n. What happens to S n as n gets larger and larger, that is, as we add more and more terms? We are considering an infinite sum written as a a a,orinsummation notation, a n. n This is called an infinite series. Since we cannot add an infinite set of numbers, we need instead the notion of a limit. In one sense, calculus is the study of limits. It is beyond the scope of this book to deal with infinite series in general, but for a geometric sequence a n,we can at least get an intuitive feeling for what happens to S n as n becomes large. In Examples 8 and 9, where a n and S n n ( n), it is reasonable to assume that gets close to 0 as n becomes large. In calculus notation n lim na n 0, from which lim S n na. We say that the infinite series, n n n 6. n converges to, and we write In general, we associate each geometric sequence ar n with an infinite geometric series ar n a ar ar ar n. n
8 458 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers The only meaning we give to this infinite sum is the limit of the sequence of partial sums, lim S n lim na na a r n, r which depends on lim na r n. Looking at different values of r, we conclude that if r is any number between and, then lim na r n 0, from which a r n lim S n lim na na r a r. Infinite geometric series Associated with every geometric sequence ar n is an infinite geometric series ar n a ar ar ar n. n If r, then the series converges to a, and we write r ar n a ar ar ar n n a r. (5) If r, then the infinite series does not have a sum, and it diverges. Repeating decimals. In Section. we said that the decimal representation of any rational number is a repeating decimal. The following example illustrates how we can use an infinite geometric series to express a repeating decimal as a fraction of integers. EXAMPLE 0 Repeating decimal Write in terms of an infinite geometric series, then use Equation (5) to express.45 in the form p, where p and q are integers. q The terms following 45 0 form an infinite geometric series with a and 000 r Since r is between and, we may use Equation (5) to express the sum as Therefore, 7 0 and.45 represent the same number. Functions represented by infinite series. The infinite series x x is geometric (with a andr x), so if x is any number between and, the
9 8. Airthmetic and Geometric Sequences 459 series converges: x x x x. Hence the function f x, where x, can be represented by the x infinite series x x. An important topic arises in calculus when we represent functions by infinite series. For instance, it can be shown that the function F x sin x is also given by sin x x x! x 5 5! x 7 7!. The representation for sin x is not a geometric series, but it does converge for every real number x. Itfollowsthatsinxcan be approximated by polynomial functions consisting of the first few terms of the infinite series. For example, if we let p x be the sum of the first four terms, p x x x! x 5 5! x 7, then p x sin x. 7! Evaluating at x 0.5, sin 0.5 p To see how good this approximation is, use your calculator to evaluate sin 0.5 (in radian mode). In fact, your calculator is probably designed to use polynomial approximations to evaluate most of its builtin functions. Following are series representations for some important functions we have studied in Chapters 4 and 5. sin x x x! x 5 5! x 7 cosx x 7!! x 4 4! x 6 6! e x x x! x! e x x x! x! EXERCISES 8. Check Your Understanding Exercises 6 True or False. Give reasons.. If a n is an arithmetic sequence, then a 6 a a 8 a 5.. The sequence beginning,,, 4 6 8,...could be an arithmetic sequence.. If c n is a geometric sequence, then c 5 r. c 4. The sequences a n and b n given by a n n and b n log 00 n are identical. 5. In a geometric sequence if the common ratio is negative, then after a certain point in the sequence, all the termswillbenegative. 6. In an arithmetic sequence if the common difference is negative, then after a certain point in the sequence, all the terms will be negative. Exercises 7 0 Fill in the blank so that the resulting statement is true k. k k. k
10 460 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers Develop Mastery Exercises 0 Arithmetic Sequences The first three terms of an arithmetic sequence are given. Find (a) the common difference, (b) the sixth and tenth terms, and (c) the sum of the first ten terms..,6,9,.... 8,, 4,.... 4,, 6,... 4.,, 5, , 5, , 0., 0.40,... 7.,,, ,, 5, ln,ln4,ln8, ln e, 4,7,... Exercises 0 Geometric Sequences The first three terms of a geometric sequence are given. Find (a) the common ratio, (b) the sixth and eighth terms, and (c) the sum of the first five terms..,,,... 9.,,,.... 8,6,,... 4., 0.5, 0.5,... 5.,,,... 6., 7.,.5, 0.75, ,,, ,,,... 0.,,,... Exercises 8 Arithmetic or Geometric? The first three terms of a sequence are given. Determine whether the sequence could be arithmetic, geometric, or neither. If arithmetic, find the common difference; if geometric, give the common ratio..,, 4,...., 4, 8,....,,, ln, ln, ln,... 6.,4,9, , 0.00, , e, e, e,... Exercises 9 6 Arithmetic Sequences Assume that the given information refers to an arithmetic sequence. Find the indicated quantities. 9. a 5, a 6 0; d, a 0. a 5, d ; a, a 0. a, a 8 5; d, S 8. a 8, a 9 ; S 4, S 6. a 8 5, S 8 64; a, S 4 4. a 6, S 6 8; a, S 6 5. a 5, d, a 4, a 6, S 6 6. a 5, a 8 4 ; a, a, S Exercises 7 44 Geometric Sequences Assume that the given information refers to a geometric sequence. Determine the indicated quantities. 7. a 4, a 6; r, a 6 8. a, a ; a 4, a 7 9. a 5 4, r ; a, S a 4 6, a 7 48; r, a 0 4. a 8, S 4; a 5, S 5 4. a 4, a 7 ; r, S a 8,a 5 0 ; a 80, S a, S ; a 6, S 6 Exercises Find x Determine the value(s) of x for which the given expressions will form the first three terms of the indicated type of sequence. 45., x, x ; arithmetic 46. x, x, x 6; arithmetic 47., x, x ; geometric 48. x, x, x 6; geometric 49. x, x, x ; arithmetic 50., x, x 4 ; geometric Exercises 5 56 Arithmetic or Geometric? Three expressions are given. Determine whether, for every real number x, they are the first three terms of an arithmetic sequence or a geometric sequence. 5. x, x, x 5 5. x, x,4 x 5. x, x, x 54. x, x, x 55. x, x, x 56. x, x, 4 x Exercises Infinite Series For the infinite series, (a) write out the first four terms, find the common ratio and aformulafors n.(b) Find the sum of the series; that is, find the limit of S n asngetslarge n n n n 4 5 n 59. n 60. n 4 n n Exercises 6 6 Sum of an Infinite Series Find the sum of the infinite geometric series Exercises6 64 Geometric Sequence, Partial Sums, Convergence The first three terms of a geometric sequence are given. (a) Use Equation (4) to find a formula for S n as a function of n, and draw a graph of S n. (b) Using Equation (5)findthelimitLof S n.(c) Trace to find the smallest valueofnforwhich S n L is less than 0.00, See Example ,.4,.44, ,., 0.96,...
11 8.4 Patterns, Guesses, and Formulas 46 Exercises Express as a quotient of two integers in reduced form. 65. (a).4, (b) (a).45, (b) (a).5, (b) (a) 0.7, (b) Evaluate the sum (a) How many integers between 00 and 000 are divisible by? (b) What is their sum? 7. Find the sum of all odd positive integers less than Find the sum of all positive integers between 400 and 500 that are divisible by. 7. If a b c and a, b, andcare the first three terms of a geometric sequence, show that the numbers log a 4, log b 4,and are three consecutive terms log c 4 of an arithmetic sequence. (Hint: Use the change of the base formula from page 8.) 74. In a geometric sequence a n of positive terms, a a and a 5 a 4 4. Find the first five terms of the sequence. 75. If the sum of the first 60 odd positive integers is subtractedfromthesumofthefirst60evenpositiveintegers, what is the result? 76. The measures of the four interior angles of a quadrilateral form four terms of an arithmetic sequence. If the smallest angle is 7, what is the largest angle? 77. In a right triangle with legs a, b, and hypotenuse c, suppose that a, b, andcare three consecutive terms of a geometric sequence. Find the common ratio r. 78. Theseatsinatheaterarearrangedinrowswith40 seats in the first row, 4 in the second, 44 in the third andsoon. (a) Howmanyseatsareinthethirtyfirstrow?Inthe middle row? (b) How many seats are in the theater? 79. A rubber ball is dropped from the top of the Washington Monument, which is 70 meters high. Suppose each time it hits the ground it rebounds of the distance of the preceding fall. (a) What total distance does the ball travel up to the instantwhenithitsthegroundforthethirdtime? (b) What total distance does it travel before it essentially comes to rest? 80. Suppose we wish to create a vacuum in a tank that contains 000 cubic feet of air. Each stroke of the vacuumpumpremoveshalfoftheairthatremainsinthe tank. (a) How much air remains in the tank after the fourth stroke? (b) How much air was removed during the fourth stroke? (c) How many strokes of the pump are required to remove at least 99 percent of the air? 8. From a helicopter hovering at 6400 feet above ground level an object is dropped. The distance s it falls in t seconds after being dropped is given by the formula s f t 6t. (a) How far does the object fall during the first second? (b) Let a n denote the distance that the object falls during the nth second, that is, a n f n f n. Find a formula for a n. What kind of sequence is a n? (c) Evaluate the sum a a a, and then find s when t. Compare these two numbers. (d) Clearly this is a finite sequence since the object cannot fall more than 6400 feet. How many terms are there in the sequence? What is the sum of these terms? 8.4 PATTERNS, GUESSES, AND FORMULAS What humans do with the language of mathematics is to describe patterns. Mathematics is an exploratory science that seeks to understand every kind of pattern patterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns. Lynn Arthur Steen Arithmetic and geometric sequences are highly structured, and it is precisely because we can analyze the regularity of their patterns that we can do so much with them. The formulas developed in the preceding section are examples of what can be done when patterns are recognized and used appropriately.
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