Counting methods and Probability theory

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13 Sequences, Probability and Counting Theory Figure 1 (credit: Robert S. donovan, Flickr.) ChAPTeR OUTl Ine 13.1 Sequences and Their notations 13.2 Arithmetic Sequences 13.3 geometric Sequences 13.4 Series and Their notations 13.5 Counting Principles 13.6 binomial Theorem 13.7 Probability Introduction A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a villa in Saint Barthélemy? A luxury convertible? A cruise around the world? e l Th ikelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over the next 30 years or so. This decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There are also personal reasons to consider when making the choice, and one can make many arguments for either decision. However, most lottery winners opt for the lump sum. In this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning. 10551056 CHAPTER 13 s qeue Ncse , Probabilit ay Nd c ou Nti Ng t hoe yr l eARnIng Obje CTIveS In this section, you will: • Write the terms of a sequence defined by an explicit formula. • Write the terms of a sequence defined by a recursive formula. • Use factorial notation. 13.1 SeQUen CeS And The IR nOTATIO n S A video game company launches an exciting new advertising campaign. They predict the number of online visits to their website, or hits, will double each day. The model they are using shows 2 hits the first day, 4 hits the second day, 8 hits the third day, and so on. See Table 1. Day 1 2 3 4 5 … Hits 2 4 8 16 32 … Table 1 If their model continues, how many hits will there be at the end of the month? To answer this question, we’ll first need to know how to determine a list of numbers written in a specic o fi rder. In this section, we will explore these kinds of ordered lists. Writing the Terms of a Sequence den fi ed by an explicit Formula One way to describe an ordered list of numbers is as a sequence. A sequence is a function whose domain is a subset of the counting numbers. The sequence established by the number of hits on the website is 2, 4, 8, 16, 32, …. The ellipsis (…) indicates that the sequence continues indefinitely. Each number in the sequence is called a term . The first five terms of this sequence are 2, 4, 8, 16, and 32. Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more ec ffi ient way to determine a specic t fi erm is by writing a formula to define the sequence. One type of formula is an explicit formula, which defines the terms of a sequence using their position in the sequence. Explicit formulas are helpful if we want to find a specic t fi erm of a sequence without finding all of the previous terms. We can use the formula to find the n th term of the sequence, where n is any positive number. In our example, each number in the sequence is double the previous number, so we can use powers of 2 to write a formula for the nth term. 2, 4, 8, 16, 22, …, ?, … ↓ ↓ ↓ ↓ ↓ ↓ 1 2 3 4 5 n 2 , 2 , 2 , 2 , 2 , …, 2 , … 1 2 3 e fi Th rst term of the sequence is 2 = 2, the second term is 2 = 4, the third term is 2 = 8, and so on. The n th term of the sequence can be found by raising 2 to the nth power. An explicit formula for a sequence is named by a lower case letter a, b, c... with the subscript n . The explicit formula for this sequence is n a = 2 . n Now that we have a formula for the nth term of the sequence, we can answer the question posed at the beginning of this section. We were asked to find the number of hits at the end of the month, which we will take to be 31 days. To st find the number of hits on the last day of the month, we need to find the 31 term of the sequence. We will substitute 31 for n in the formula. 31 a = 2 31 = 2,147,483,648 SECTION 13.1 s qeu e Ncse a N d t hei N r tioato Ns 1057 If the doubling trend continues, the company will get 2,147,483,648 hits on the last day of the month. That is over 2.1 billion hits The huge number is probably a little unrealistic because it does not take consumer interest and competition into account. It does, however, give the company a starting point from which to consider business decisions. Another way to represent the sequence is by using a table. The first five terms of the sequence and the n th term of the sequence are shown in Table 2. n 1 2 3 4 5 n n nth term of the sequence, a 2 4 8 16 32 2 n Table 2 Graphing provides a visual representation of the sequence as a set of distinct points. We can see from the graph in Figure 1 that the number of hits is rising at an exponential rate. This particular sequence forms an exponential function. a n 36 32 (5, 32) 28 24 20 16 (4, 16) 12 (3, 8) 8 (2, 4) 4 (1, 2) n 0 1 23456 Figure 1 Lastly, we can write this particular sequence as n, 2, 4, 8, 16, 32, …, 2 … A sequence that continues indefinitely is called an infinite sequence . The domain of an infinite sequence is the set of counting numbers. If we consider only the first 10 terms of the sequence, we could write n 2, 4, 8, 16, 32, …, 2 , … , 1024. This sequence is called a finite sequence because it does not continue indefinitely. sequence A sequence is a function whose domain is the set of positive integers. A finite sequence is a sequence whose domain consists of only the first n positive integers. The numbers in a sequence are called terms . The variable a with a number subscript is used to represent the terms in a sequence and to indicate the position of the term in the sequence. a , a , a , … , a , … 1 2 3 n We call a the first term of the sequence, a the second term of the sequence, a the third term of the sequence, 1 2 3 and so on. e Th term an is called the nth term of the sequence, or the general term of the sequence. An explicit formula defines the n th term of a sequence using the position of the term. A sequence that continues indefinitely is an infinite sequence . Q & A… Does a sequence always have to begin with a ? 1 No. In certain problems, it may be useful to define the initial term as a instead of a . In these problems, the domain 0 1 of the function includes 0. 1058 CHAPTER 13 s qeue Ncse , Probability a N d c ou Nti N g t hoe yr How To… Given an explicit formula, write the first n terms of a sequence. 1. Substitute each value of n into the formula. Begin with n = 1 to find the first term, a . 1 2. To find the second term, a , use n = 2. 2 3. Continue in the same manner until you have identie fi d all n terms. Example 1 Writing the Terms of a Sequence Defined by an Explicit Formula Write the first five terms of the sequence defined by the explicit formula a = −3n + 8. n Solution Substitute n = 1 into the formula. Repeat with values 2 through 5 for n. n = 1 a = −3(1) + 8 = 5 1 n = 2 a = −3(2) + 8 = 2 2 n = 3 a = −3(3) + 8 = −1 3 n = 4 a = −3(4) + 8 = −4 4 n = 5 a = −3(5) + 8 = −7 5 e fi Th r st five terms are 5, 2, − 1, −4, −7. Analysis The sequence values can be listed in a table. A table, such as Table 3 , is a convenient way to input the function into a graphing utility. n 1 2 3 4 5 a 5 2 −1 −4 −7 n Table 3 A graph can be made from this table of values. From the graph in Figure 2, we can see that this sequence represents a linear function, but notice the graph is not continuous because the domain is over the positive integers only. a n 6 5 (1, 5) 4 3 2 (2, 2) 1 n 0 –1 1 23 45 6 (3, –1) –1 –2 –3 –4 (4, –4) –5 –6 (5, –7) –7 –8 Figure 2 Try It 1 Write the first five terms of the sequence defined by the explicit formula t = 5n − 4. nSECTION 13.1 s qeu e Ncse a N d t hei r Ntioato Ns 1059 Investigating Alternating Sequences Sometimes sequences have terms that are alternate. In fact, the terms may actually alternate in sign. The steps to finding terms of the sequence are the same as if the signs did not alternate. However, the resulting terms will not show increase or decrease as n increases. Let’s take a look at the following sequence. 2, −4, 6, −8 Notice the first term is greater than the second term, the second term is less than the third term, and the third term is greater than the fourth term. This trend continues forever. Do not rearrange the terms in numerical order to interpret the sequence. How To… Given an explicit formula with alternating terms, write the first n terms of a sequence. 1. Substitute each value of n into the formula. Begin with n = 1 to find the first term, a . The sign of the term is given 1 n by the (−1) in the explicit formula. 2. To find the second term, a , use n = 2. 2 3. Continue in the same manner until you have identie fi d all n terms. Example 2 Writing the Terms of an Alternating Sequence Defined by an Explicit Formula Write the first five terms of the sequence. n 2 (−1) n _ a = n n + 1 Solution Substitute n = 1, n = 2, and so on in the formula. 1 1 (−1) 1 1 _______ __ n = 1 a = = − 1 1 + 1 2 2 2 (−1) 2 4 _______ __ n = 2 a = = 2 2 + 1 3 3 2 (−1) 3 9 _______ __ n = 3 a = = − 3 3 + 1 4 4 2 (−1) 4 16 _______ __ n = 4 a = = 4 4 + 1 5 5 2 (−1) 5 25 _______ ___ n = 5 a = = − 5 5 + 1 6 1 4 9 16 25 __ __ __ ___ ___ e fi Th r st five terms are − , , − , , −   2 3 4 5 6 Analysis The graph of this function, shown in Figure 3 , looks different from the ones we have seen previously in this section because the terms of the sequence alternate between positive and negative values. a n 4 1 4, 3 3 5 2 1 2, 1 1 3 n 0 –1 1 23 45 6 –1 1 1, − 2 –2 1 3, −2 4 –3 –4 1 5, −4 6 –5 Figure 3 1060 CHAPTER 13 s qeue Ncse , Probability a N d c ou Nti N g t hoe yr Q & A… In Example 2, does the (−1) to the power of n account for the oscillations of signs? Yes, the power might be n, n + 1, n − 1, and so on, but any odd powers will result in a negative term, and any even power will result in a positive term. Try It 2 4n _____ Write the first five terms of the sequence. a = n n (−2) Investigating Piecewise Explicit Formulas We’ve learned that sequences are functions whose domain is over the positive integers. This is true for other types of functions, including some piecewise functions. Recall that a piecewise function is a function defined by multiple subsections. A die ff rent formula might represent each individual subsection. How To… Given an explicit formula for a piecewise function, write the first n terms of a sequence 1. Identify the formula to which n = 1 applies. 2. To find the first term, a , use n = 1 in the appropriate formula. 1 3. Identify the formula to which n = 2 applies. 4. To find the second term, a , use n = 2 in the appropriate formula. 2 5. Continue in the same manner until you have identie fi d all n terms. Example 3 Writing the Terms of a Sequence Defined by a Piecewise Explicit Formula Write the first six terms of the sequence. 2 n if n is not divisible by 3 a = n __ n if n is divisible by 3 3 2 Solution Substitute n = 1, n = 2, and so on in the appropriate formula. Use n when n is not a multiple of 3. Use n _ when n is a multiple of 3. 3 2 2 a = 1 = 1 1 is not a multiple of 3. Use n . 1 2 2 a = 2 = 4 2 is not a multiple of 3. Use n . 2 3 n __ __ a = = 1 3 is a multiple of 3. Use . 3 3 3 2 2 a = 4 = 16 4 is not a multiple of 3. Use n . 4 2 2 a = 5 = 25 5 is not a multiple of 3. Use n . 5 6 n __ __ a = = 2 6 is a multiple of 3. Use . 6 3 3 e fi Th rst six terms are 1, 4, 1, 16, 25, 2. Analysis Every third point on the graph shown in Figure 4 stands out from the two nearby points. This occurs because the sequence was den fi ed by a piecewise function. a n 28 24 20 16 12 8 4 n 0 1 23 4 56 7 Figure 4 SECTION 13.1 s qeu e Ncse a N d t hei N r tioato Ns 1061 Try It 3 3 2n if n is odd Write the first six terms of the sequence. a = n 5n ___ if n is even 2 Finding an Explicit Formula u Th s far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the explicit formula for the nth term of a sequence is not given. Instead, we are given several terms from the sequence. When this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to finding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases. How To… Given the first few terms of a sequence, find an explicit formula for the sequence. 1. Look for a pattern among the terms. 2. If the terms are fractions, look for a separate pattern among the numerators and denominators. 3. Look for a pattern among the signs of the terms. 4. Write a formula for a in terms of n. Test your formula for n = 1, n = 2, and n = 3. n Example 4 Writing an Explicit Formula for the nth Term of a Sequence Write an explicit formula for the nth term of each sequence. 2 3 4 5 6 ___ ___ ___ ___ ___ a. − , , − , , − , …   11 13 15 17 19 2 2 2 2 2 ___ ___ ___ _____ ______ b. − , − , − , − , − , …   25 125 625 3,125 15,625 4 5 6 7 8 c. e , e , e , e , e , … Solution Look for the pattern in each sequence. n a. e t Th er ms alternate between positive and negative. We can use (−1) to make the terms alternate. The numerator can be represented by n + 1. The den ominator can be represented by 2n + 9. n (−1) (n + 1) ___________ a = n 2n + 9 b. e t Th er ms are all negative. 2 2 2 2 2 ___ ___ ___ _____ ______ − , − , − , − , − , … e n Th umerator is 2   25 125 625 3,125 15,625 2 2 2 2 2 2 2 __ __ __ __ __ __ __ − , − , − , − , − , − , … − e de Th nominators are increasing powers of 5   2 3 4 5 6 7 n 5 5 5 5 5 5 5 n + 1 So we know that the fraction is negative, the numerator is 2, and the denominator can be represented by 5 . 2 ______ a = − n n + 1 5 4 c. e t Th er ms are powers of e. For n = 1, the first term is e so the exponent must be n + 3. n + 3 a = e n Try It 4 Write an explicit formula for the nth term of the sequence. 9, −81, 729, −6,561, 59,049, … 1062 CHAPTER 13 s qeue Ncse , Probabilit ay N d c ou Nti Ng t hoe yr Try It 5 Write an explicit formula for the nth term of the sequence. 3 9 27 81 243 __ __ ___ ___ ___ − , − , − , − , − , ...   4 8 12 16 20 Try It 6 Write an explicit formula for the nth term of the sequence. 1 1 __ _ 2 , , 1, e, e , ...   2 e e Writing the Terms of a Sequence den fi ed by a Recursive Formula Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and die ff rent varieties of daisies that may have 21 or 34 petals. Each term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a recursive formula, a formula that defines the terms of a sequence using previous terms. A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining a in terms n of preceding terms. For example, suppose we know the following: a = 3 1 a = 2a − 1, for n ≥ 2 n n − 1 We can find the subsequent terms of the sequence using the first term. a = 3 1 a = 2a − 1 = 2(3) − 1 = 5 2 1 a = 2a − 1 = 2(5) − 1 = 9 3 2 a = 2a − 1 = 2(9) − 1 = 17 4 3 So the first four terms of the sequence are 3, 5, 9, 17 . The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms. a = 1 1 a = 1 2 a = a + a for n ≥ 3 n n − 1 n − 2 To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told previously that the eighth and ninth terms are 21 and 34, so a = a + a = 34 + 21 = 55 10 9 8 recursive formula A recursive formula is a formula that defines each term of a sequence using preceding term(s ). Recursive formulas must always state the initial term, or terms, of the sequence.SECTION 13.1 s qeu e Ncse a N d t hei N r tioato Ns 1063 Q & A… Must the first two terms always be given in a recursive formula? No. The Fibonacci sequence defines each term using the two preceding terms, but many recursive formulas define each term using only one preceding term. These sequences need only the first term to be defined. How To… Given a recursive formula with only the first term provided, write the first n terms of a sequence. 1. Identify the initial term, a , which is given as part of the formula. This is the first term. 1 2. To find the second term, a , substitute the initial term into the formula for a . Solve. 2 n − 1 3. To find the third term, a , substitute the second term into the formula. Solve. 3 4. Repeat until you have solved for the nth term. Example 5 Writing the Terms of a Sequence Defined by a Recursive Formula Write the first five terms of the sequence defined by the recursive formula. a = 9 1 a = 3a − 20, for n ≥ 2 n n − 1 Solution The first term is given in the formula. For each subsequent term, we replace a with the value of the n − 1 preceding term. n = 1 a = 9 1 n = 2 a = 3a − 20 = 3(9) − 20 = 27 − 20 = 7 2 1 n = 3 a = 3a − 20 = 3(7) − 20 = 21 − 20 = 1 3 2 n = 4 a = 3a − 20 = 3(1) − 20 = 3 − 20 = −17 4 3 n = 5 a = 3a − 20 = 3( −17) − 20 = −51 − 20 = −71 5 4 e fi Th rst five terms are 9, 7, 1, – 17, – 71. See Figure 5. a n 20 (1, 9) (2, 7) (3, 1) n 0 −1 12 34 56 (4, −17) −20 −40 −60 (5, −71) −80 Figure 5 Try It 7 Write the first five terms of the sequence defined by the recursive formula. a = 2 1 a = 2a + 1, for n ≥ 2 n n − 1 How To… Given a recursive formula with two initial terms, write the first n terms of a sequence. 1. Identify the initial term, a , which is given as part of the formula. 1 2. Identify the second term, a , which is given as part of the formula. 2 3. To find the third term, substitute the initial term and the second term into the formula. Evaluate. 4. Repeat until you have evaluated the nth term.1064 CHAPTER 13 s qeue Ncse , Probabilit ay Nd c ou Nti Ng t hoe yr Example 6 Writing the Terms of a Sequence Defined by a Recursive Formula Write the first six terms of the sequence defined by the recursive formula. a = 1 1 a = 2 2 a = 3a + 4a , for n ≥ 3 n n − 1 n − 2 Solution e fi Th rst two terms are given. For each subsequent term, we replace a and a with the values of the n − 1 n − 2 two preceding terms. n = 3 a = 3a + 4a = 3(2) + 4(1) = 10 3 2 1 n = 4 a = 3a + 4a = 3(10) + 4(2) = 38 4 3 2 n = 5 a = 3a + 4a = 3(38) + 4(10) = 154 5 4 3 n = 6 a = 3a + 4a = 3(154) + 4(38) = 614 6 5 4 Th e first six terms are 1, 2, 10, 38, 154, 614. See Figure 6 . a n 700 (6, 614) 600 500 400 300 200 (5, 154) 100 (2, 2) (1, 1) (3, 10) (4, 38) n 0 123 45 67 Figure 6 Try It 8 Write the first 8 terms of the sequence defined by the recursive formula. a = 0 1 a = 1 2 a = 1 3 a n − 1 _____ a = + a , for n ≥ 4 n a n − 3 n − 2 Using Factorial notation e Th formulas for some sequences include products of consecutive positive integers. n factorial, written as n, is the product of the positive integers from 1 to n. For example, 4 = 4 ⋅ 3 2 ⋅ ⋅ 1 = 24 5 = 5 ⋅ 4 3 ⋅ 2 ⋅ ⋅ 1 = 120 An example of formula containing a factorial is a = (n + 1). The sixth term of the sequence can be found by n substituting 6 for n. a = (6 + 1) = 7 = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5,040 6 e f Th ac torial of any whole number n is n(n − 1) We can therefore also think of 5 as 5 4 ⋅ .SECTION 13.1 s qeu e Ncse a Nd t hei N r tioato Ns 1065 factorial n factorial is a mathematical operation that can be defined using a recursive formula. The factorial of n , denoted n, is defined for a positive integer n as: 0 = 1 1 = 1 n = n(n − 1)(n − 2) ⋯ (2)(1), for n ≥ 2 e s Th pecial case 0 is defined as 0 = 1. Q & A… Can factorials always be found using a calculator? No. Factorials get large very quickly—faster than even exponential functions When the output gets too large for the calculator, it will not be able to calculate the factorial. Example 7 Writing the Terms of a Sequence Using Factorials 5n _______ Write the first five terms of the sequence defined by the explicit formula a = . n (n + 2) Solution Substitute n = 1, n = 2, and so on in the formula. 5(1) 5 5 5 _______ __ ______ __ n = 1 a = = = = 1 (1 + 2) 3 3 · 2 · 1 6 5(2) 10 10 5 _______ ___ _________ ___ n = 2 a = = = = 2 (2 + 2) 4 4 · 3 · 2 · 1 12 5(3) 15 15 1 _______ ___ ___________ __ n = 3 a = = = = 3 (3 + 2) 5 5 · 4 · 3 · 2 · 1 8 5(4) 20 20 1 _______ ___ _____________ ___ n = 4 a = = = = 4 (4 + 2) 6 6 · 5 · 4 · 3 · 2 · 1 36 5(5) 25 25 5 _______ ___ _______________ _____ n = 5 a = = = = 5 (5 + 2) 7 7 · 6 · 5 · 4 · 3 · 2 · 1 1,008 5 5 1 1 5 __ ___ __ ___ _____ e fi Th r st five terms are , , , , .   6 12 8 36 1,008 Analysis Figure 7 shows the graph of the sequence. Notice that, since factorials grow very quickly, the presence of the factorial term in the denominator results in the denominator becoming much larger than the numerator as n increases. This means the quotient gets smaller and, as the plot of the terms shows, the terms are decreasing and nearing zero. a n 1 5 5 1, 6 6 4 6 3 6 5 2, 12 2 6 1 3, 1 8 1 5 6 4, 5, 36 1008 n 0 1 23 45 6 Figure 7 Try It 9 (n + 1) _______ Write the first five terms of the sequence defined by the explicit formula a = . n 2n Access this online resource for additional instruction and practice with sequences. • Finding Terms in a Sequence (http://openstaxcollege.org/l/findingterms) 1066 CHAPTER 13 s qeue Ncse , Probabilit ay N d c ou Nti N g t hoe yr 13.1 SeCTIOn exeRCISeS veRbAl 1. Discuss the meaning of a sequence. If a finite 2. Describe three ways that a sequence can be defined. sequence is defined by a formula, what is its domain? What about an infinite sequence? 3. Is the ordered set of even numbers an infinite 4. What happens to the terms a of a sequence when n sequence? What about the ordered set of odd there is a negative factor in the formula that is raised numbers? Explain why or why not. to a power that includes n? What is the term used to describe this phenomenon? 5. What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial. Algeb RAIC For the following exercises, write the first four terms of the sequence. n 16 2 n _____ n − 1 __ 6. a = 2 − 2 7. a = − 8. a = −(−5) 9. a = 3 n n n n n + 1 n 2 2n + 1 n ______ n − 1 n − 1 ______ 10. a = 11. a = 1.25 ⋅ (−4) 12. a = −4 ⋅ (−6) 13. a = n 3 n n n n 2n + 1 n − 1 n 4 ( ⋅ −5) 14. a = (−10) + 1 __________ 15. a = − n   n 5 For the following exercises, write the first eight terms of the piecewise sequence. 2 n n _ ( −2) − 2 if n is even if n ≤ 5 16. a = 17. a = 2n + 1 n n − 1 n (3) if n is odd 2 n − 5 if n 5 2 n − 1 (2n + 1) if n is divisible by 4 −0.6 ⋅ 5 if n is prime or 1 19. a = 18. a = n 2 n − 1 _ n 2.5 ( ⋅ −2) if n is composite if n is not divisible by 4 n 2 4(n − 2) if n ≤ 3 or n 6 2 20. a = n − 2 _ n if 3 n ≤ 6 4 For the following exercises, write an explicit formula for each sequence. 4 16 __ ___ 21. 4, 7, 12, 19, 28, … 22. −4, 2, − 10, 14, − 34, … 23. 1, 1, , 2, , … 3 5 3 4 1 2 1 − e 1 − e 1 − e 1 − e 1 1 1 1 ______ ______ _____ _____ __ __ __ ___ 24. 0, , , , , … 25. 1, − , , − , , … 2 3 4 5 1 + e 1 + e 2 4 8 16 1 + e 1 + e For the following exercises, write the first five terms of the sequence. 26. a = 9, a = a + n 27. a = 3, a = (−3)a 1 n n − 1 1 n n − 1 n − 1 a + 2n (−3) n − 1 _ _ 28. a = −4, a = 29. a = −1, a = 1 n 1 n a − 1 a − 2 n − 1 n − 1 n 1 __ 30. a = −30, a = 2 + a ( ) 1 n n − 1   2 For the following exercises, write the first eight terms of the sequence. 1 ___ 31. a = , a = 1, a = 2a 3a 32. a = −1, a = 5, a = a 3 − a ( )( ) ( ) 1 2 n n − 2 n − 1 1 2 n n − 2 n −1 24 2 a + 2 ( ) n −1 __ 33. a = 2, a = 10, a = 1 2 n a n − 2SECTION 13.1 s ectio N e xercises 1067 For the following exercises, write a recursive formula for each sequence. 34. −2.5, − 5, − 10, − 20, − 40, … 35. −8, − 6, − 3, 1, 6, … 36. 2, 4, 12, 48, 240, … 3 3 3 __ ___ ___ 37. 35, 38, 41, 44, 47, … 38. 15, 3, , , , … 5 25 125 For the following exercises, evaluate the factorial. 12 100 ___ ____ 12 ___ 39. 6 41. 42. 40.   6 99 6 For the following exercises, write the first four terms of the sequence. n 3 ⋅ n n 100 ⋅ n __ _____ _________ ________ 43. a = 44. a = 45. a = 46. a = n 2 n n 2 n n 4 ⋅ n n − n − 1 n(n − 1) gRAPhICAl For the following exercises, graph the first five terms of the indicated sequence n (−1) 4 + n _____ _ 2 47. a = + n 49. a = 2, a = (−a + 1) if n in even n 1 n n − 1 n 48. a = 2n n 3 + n if n is odd (n + 1) _______ 50. a = 1, a = a + 8 51. a = n n n − 1 n (n − 1) For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph. 52. 53. 54. a n a n a n 8 (5, 8) 18 15 7 13 (5, 13) 15 (1, 12) 6 11 (4, 11) 12 (2, 9) 5 9 (3, 9) 9 (3, 6) (2, 7) (4, 4) 7 4 6 (4, 3) 5 (1, 5) 3 3 (5, 0) 3 2 (3, 2) n (2, 1) 0 12 34 56 7 1 n (1, 0.5) 0 12 34 56 7 n 0 1 23 45 6 7 For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph. 55. 56. a a n n 16 (1, 16) 22 (5, 21) 20 12 16 8 (2, 8) (4, 13) 12 4 (3, 4) (3, 9) (4, 2) 8 (5, 1) (2, 7) n 0 12 34 5 (1, 6) 4 n 0 12 34 51068 CHAPTER 13 s qeue Ncse , Probability a Nd c ou Nti Ng t hoe yr TeChn Ol Ogy Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term a and press ENTER. 1 • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes 2ND ANS for the previous term a . Press ENTER. n − 1 • Continue pressing ENTER to calculate the values for each successive term. For the following exercises, use the steps above to find the indicated term or terms for the sequence. 87 ___ th 57. Find the first five terms of the sequence a = , 58. Find the 15 term of the sequence 1 111 4 12 __ ___ a = 625, a = 0.8a + 18. a = a + . U se the Frac feature to give 1 n n − 1 n n − 1 3 37 fractional results. 59. Find the first five terms of the sequence 60. Find the first ten terms of the sequence (a + 1) (a ) − 1 n − 1 a = 2, a = 2 + 1. n − 1 _ 1 n a = 8, a = . 1 n a n − 1 61. Find the tenth term of the sequence a = 2, a = na 1 n n − 1 Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following. • In the home screen, press 2ND LIST. • Scroll over to OPS and choose “seq(” from the dropdown list. Press ENTER. • In the line headed “Expr:” type in the explicit formula, using the X,T, θ, n button for n • In the line headed “Variable:” type in the variable used on the previous step. • In the line headed “start:” key in the value of n that begins the sequence. • In the line headed “end:” key in the value of n that ends the sequence. • Press ENTER 3 times to return to the home screen. You will see the sequence syntax on the screen. Press ENTER to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press 2ND LIST. • Scroll over to OPS and choose “seq(” from the dropdown list. Press ENTER. • Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas. See the instructions above for the description of each item. • Press ENTER to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 62. List the first five terms of the sequence. 63. List the first six terms of the sequence. 3 2 28 5 _ ___ n − 3.5n + 4.1n − 1.5 ___________________ a = − n + a = n n 9 3 2.4n 64. List the first five terms of the sequence. 65. List the first four terms of the sequence. n − 1 n 15n ( ⋅ −2) ____________ a = 5.7 + 0.275(n − 1) a = n n 47 n __ 66. List the first six terms of the sequence a = . n n exTen SIOnS 2 n + 4n + 4 __________ 67. Consider the sequence defined by a = −6 − 8n. Is 68. What term in the sequence a = has n n 2(n + 2) a = −421 a term in the sequence? Verify the result. the value 41? Verify the result. n 69. Find a recursive formula for the sequence 70. Calculate the first eight terms of the sequences 1, 0, −1, −1, 0, 1, 1, 0, −1, −1, 0, 1, 1, ... (n + 2) _______ 3 2 a = and b = n + 3n + 2n, and then n n (n − 1) (Hint: find a pattern for an based on the first two make a conjecture about the relationship between terms.) these two sequences. 71. Prove the conjecture made in the preceding exercise. SECTION 13.2 a rit hmetic s qeue Ncse 1069 l eARnIng Obje CTIveS In this section, you will: • Find the common difference for an arithmetic sequence. • Write terms of an arithmetic sequence. • Use a recursive formula for an arithmetic sequence. • Use an explicit formula for an arithmetic sequence. 13. 2 ARIThmeTIC SeQUen CeS Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year. As an example, consider a woman who starts a small contracting business. She purchases a new truck for 25,000. Ae ft r five years, she estimates that she will be able to sell the truck for 8,000. e Th loss in value of the truck will therefore be 17,000, which is 3,400 per year for five years. The truck will be worth 21,600 after the first year; 18,200 after two years; 14,800 after three years; 11,400 after four years; and 8,000 at the end of five years. In this section, we will consider specic k fi inds of sequences that will allow us to calculate depreciation, such as the truck’s value. Finding Common differences The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common die ff rence of the sequence. For this sequence, the common die ff rence is − 3,400. −3,400 −3,400 −3,400 −3,400 −3,400 25000, 21600, 18200, 14800, 11400, 8000 e s Th equence below is another example of an arithmetic sequence. In this case, the constant die ff rence is 3. You can choose any term of the sequence, and add 3 to find the subsequent term. +3 +3 +3 +3 3, 6, 9, 12, 15, ... arithmetic sequence An arithmetic sequence is a sequence that has the property that the die ff rence between any two consecutive terms is a constant. This constant is called the common die ff rence . If a is the first term of an arithmetic sequence and 1 d is the common die ff rence, the sequence will be: a = a , a + d, a + 2d, a + 3d,... n 1 1 1 1 Example 1 Finding Common Differences Is each sequence arithmetic? If so, find the common die ff rence. a. 1, 2, 4, 8, 16, ... b. −3, 1, 5, 9, 13, ... Solution Subtract each term from the subsequent term to determine whether a common die ff rence exists. a. e s Th e quence is not arithmetic because there is no common difference. 2 − 1 = 1 4 − 2 = 2 8 − 4 = 4 16 − 8 = 8 b. e s Th e quence is arithmetic because there is a common difference. The common difference is 4. 1 − (−3) = 4 5 − 1 = 4 9 − 5 = 4 13 − 9 = 41070 CHAPTER 13 s qeue Ncse , Probabilit ay Nd c ou Nti N g t hoe yr Analysis e g Th raph of each of these sequences is shown in Figure 1 . We can see from the graphs that, although both sequences show growth, a is not linear whereas b is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line. a a n n 20 20 16 16 12 12 8 8 4 4 n n 0 0 1 23 45 6 1 2 34 56 −4 −4 (a) (b) Figure 1 Q & A… If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common die ff rence? No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common die ff rence. Try It 1 Is the given sequence arithmetic? If so, find the common die ff rence. 18, 16, 14, 12, 10, … Try It 2 Is the given sequence arithmetic? If so, find the common die ff rence. 1, 3, 6, 10, 15, … Writing Terms of Arithmetic Sequences Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of n and d into formula below. a = a + (n − 1)d n 1 How To… Given the first term and the common die ff rence of an arithmetic sequence, find the first several terms. 1. Add the common die ff rence to the first term to find the second term. 2. Add the common die ff rence to the second term to find the third term. 3. Continue until all of the desired terms are identie fi d. 4. Write the terms separated by commas within brackets. Example 2 Writing Terms of Arithmetic Sequences Write the first five terms of the arithmetic sequence with a = 17 and d = −3. 1 Solution Adding −3 is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term. e fi Th rst five terms are 17, 14, 11, 8, 5SECTION 13.2 a rit hmetic s qeue Ncse 1071 Analysis As expected, the graph of the sequence consists of points on a line as shown in Figure 2. a n 20 16 12 8 4 n 0 12 3 456 Figure 2 Try It 3 List the first five terms of the arithmetic sequence with a = 1 and d = 5. 1 How To… Given any first term and any other term in an arithmetic sequence, find a given term. 1. Substitute the values given for a , a , n into the formula a = a + (n − 1)d to solve for d. 1 n n 1 2. Find a given term by substituting the appropriate values for a , n, and d into the formula a = a + (n − 1)d. 1 n 1 Example 3 Writing Terms of Arithmetic Sequences Given a = 8 and a = 14, find a . 1 4 5 Solution e s Th equence can be written in terms of the initial term 8 and the common die ff rence d . 8, 8 + d, 8 + 2d, 8 + 3d + 3d = 8 + 3d. We know the fourth term equals 14; we know the fourth term has the form a 1 We can find the common die ff rence d . = a + (n − 1)d a n 1 a = a + 3d 4 1 = 8 + 3d Write the fourth term of the sequence in terms of a and d. a 4 1 14 = 8 + 3d Substitute 14 for a . 4 d = 2 Solve for the common difference. Find the fifth term by adding the common die ff rence to the fourth term. = a + 2 = 16 a 5 4 Analysis Notice that the common difference is added to the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to the first term nine times or by using the equation a = a + (n − 1)d. n 1 Try It 4 Given a = 7 and a = 17, find a . 3 5 2 Using Recursive Formulas for Arithmetic Sequences Some arithmetic sequences are defined in terms of the previous term using a recursive formula. e Th formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the 1072 CHAPTER 13 s qeue Ncse , Probabilit ay Nd c ou Nti N g t hoe yr common die ff rence. For example, if the common die ff rence is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given. a = a + d n ≥ 2 n n − 1 recursive formula for an arithmetic sequence e r Th ecursive formula for an arithmetic sequence with common die ff rence d is: a = a + d n ≥ 2 n n − 1 How To… Given an arithmetic sequence, write its recursive formula. 1. Subtract any term from the subsequent term to find the common die ff rence. 2. State the initial term and substitute the common die ff rence into the recursive formula for arithmetic sequences. Example 4 Writing a Recursive Formula for an Arithmetic Sequence Write a recursive formula for the arithmetic sequence. −18, −7, 4, 15, 26, … Solution e fi Th rst term is given as − 18. The common die ff rence can be found by subtracting the first term from the second term. d = −7 − (−18) = 11 Substitute the initial term and the common die ff rence into the recursive formula for arithmetic sequences. a = −18 1 a = a + 11, for n ≥ 2 n n − 1 Analysis We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure 3. The growth pattern of the sequence shows the constant difference of 11 units. a n 30 20 10 n 0 12 34 56 −10 −20 Figure 3 Q & A… Do we have to subtract the first term from the second term to find the common die ff rence? No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common die ff rence. Try It 5 Write a recursive formula for the arithmetic sequence. 25, 37, 49, 61, …SECTION 13.2 a rit hmetic s qeue Ncse 1073 Using explicit Formulas for Arithmetic Sequences We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept. a = a + d(n − 1) n 1 To find the y -intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence. −50 −50 −50 −50 200, 150, 100, 50, 0, ... e c Th ommon die ff rence is − 50, so the sequence represents a linear function with a slope of −50. To find the y -intercept, we subtract −50 from 200: 200 − ( −50) = 200 + 50 = 250. You can also find the y -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in Figure 4. a n 250 200 150 100 50 n 0 1 23 45 6 Figure 4 Recall the slope-intercept form of a line is y = mx + b. When dealing with sequences, we use a in place of y and n in n place of x. If we know the slope and vertical intercept of the function, we can substitute them for m and b in the slope- intercept form of a line. Substituting − 50 for the slope and 250 for the vertical intercept, we get the following equation: a = −50n + 250 n We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is a = 200 − 50(n − 1), which simplie fi s to a = −50n + 250. n n explicit formula for an arithmetic sequence An explicit formula for the nth term of an arithmetic sequence is given by a = a + d (n − 1) n 1 How To… Given the first several terms for an arithmetic sequence, write an explicit formula. 1. Find the common die ff rence, a − a . 2 1 2. Substitute the common die ff rence and the first term into a = a + d(n − 1). n 1 Example 5 Writing the nth Term Explicit Formula for an Arithmetic Sequence Write an explicit formula for the arithmetic sequence. 2, 12, 22, 32, 42, …1074 CHAPTER 13 s qeue Ncse , Probability a Nd c ou Nti N g t hoe yr Solution e c Th ommon die ff rence can be found by subtracting the first term from the second term. d = a − a 2 1 = 12 − 2 = 10 e c Th ommon difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify. a = 2 + 10(n − 1) n a = 10n − 8 n Analysis The graph of this sequence, represented in Figure 5 , shows a slope of 10 and a vertical intercept of −8. a n 50 40 30 20 10 n 0 1 23456 7 8 9 10 –10 Figure 5 Try It 6 Write an explicit formula for the following arithmetic sequence. 50, 47, 44, 41, … Finding the Number of Terms in a Finite Arithmetic Sequence Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common die ff rence, and then determine how many times the common die ff rence must be added to the first term to obtain the final term of the sequence. How To… Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms. 1. Find the common die ff rence d . 2. Substitute the common die ff rence and the first term into a = a + d(n − 1). n 1 3. Substitute the last term for a and solve for n. n Example 6 Finding the Number of Terms in a Finite Arithmetic Sequence Find the number of terms in the finite arithmetic sequence. 8, 1, −6, ... , −41 Solution e c Th ommon die ff rence can be found by subtracting the first term from the second term. 1 − 8 = −7

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