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All Textbook Solutions for Precalculus

For Exercises 41-46, two terms of an arithmetic sequence are given. Find the indicated term. (See Example 6) c14=7.5,c101=29.25 ; Find c400 .46PE 47PE If the third and fourth terms of an arithmetic sequence are 12 and 16 , what are the first and second terms?49PE 50PE 51PE Find the sum of the first 60 terms of the sequence, 2,10,18,26,Find the sum, 5+4.5+4+3.5++30.5 54PE 55PE 56PE 57PE For Exercises 55-64, find the sum. (See Example 8) i=1403i7 59PE For Exercises 55-64, find the sum. (See Example 8) k=1141414k 61PE 62PE 63PE 64PE 65PE Jose must choose between two job offers. The first job pays 50,000 the first year. Each year thereafter, he would receive a raise of 2400 . A second job offers 54,000 the first year with a raise of 2000 each year thereafter. However, with the second job, Jose would have to pay 100 per month out of his paycheck for health insurance. a. If Jose anticipates working for the company for 6yr , find the total amount he would earn from each job. b. If he anticipates working for the company for 12yr , find the total amount he would earn from each job.67PE 68PE The students at Prairiewood Elementary plan to make a pyramid out of plastic cups. The bottom row has 15 cups. Moving up the pyramid, the number of cups in each row decreases by 1 . a. If the students build the pyramid so that there are 12 rows, how many cups will be at the top? b. How many total cups will be required?70PE 71PE 72PE 73PE 74PE Compute the sum of the first 50 positive integers that are exactly divisible by 5 .76PE 77PE 78PE 79PE 80PE 81PE The arithmetic mean (average) to two numbers c and d is given by x=c+d2 . The value x is equidistant between c and d , so the sequence c,x,d is an arithmetic sequence. Inserting k equally values between c and d , yields the arithmetic sequence c,x1,x2,x3,x4,,xk,d . Use this information for Exercises 81-82. Insert four arithmetic means between 19 and 64.83PE 84PE Determine the nth term of the arithmetic sequence whose nth partial sum is n2+2n .86PE Determine whether the sequence is geometric. If so, identify the common ratio. a. 80,20,5,54, b. 1,8,64,640,2SP Write the nth terms of the geometric sequence. 4,5,254,12516,Find the sixth term of a geometric sequence an given that a1=64 and a2=16 .5SP 6SP Find the sum of the finite geometric series. 3+6+12++768 Find the sum if possible. a. i=1434i1 b. 2+52+258+12532+9SP 10SP Suppose that an employee contributes $100 to an annuity at the end of each month for 25yr . If the annuity earns 7%, a. Determine the value of the annuity at the end of the 25-yr period. b. How much interest will be earned?A sequence is a sequence in which each term after the first is the product of the preceding term and a fixed nonzero real number, called the common r.The nth term of a geometric sequence with first term a1 and common ratio r is given by an=.3PE The sum Sn of the first n terms of a geometric sequence with first term a1 and common ratio r is given by the formula Sn=.5PE 6PE A sequence of payments made at equal intervals over a fixed period of time is called an .Suppose that an infinite series a1+a2+a3++an approaches a value L as n. . Then the series . Otherwise, the series .For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r. (See Example 1) 6,18,54,162,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r. (See Example 1) 4,20,100,500,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r. (See Example 1) 7,72,74,78,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r. (See Example 1) 5,53,59,527,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r . (See Example 1) 3,12,60,360,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r. (See Example 1) 7,14,42,88,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r. (See Example 1) 5,5,55,25,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r. (See Example 1) 73,493,7,773,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r . (See Example 1) 2,4t,8t2,16t3,For Exercises 9-18, determine whether the sequence is geometric. If so, find the value of r . (See Example 1) 5a2,15a4,45a6,135a8,For Exercises 19-24, write the first five terms of a geometric sequence an based on the given information about the sequence. (See Example 2) a1=7andr=2 20PE 21PE For Exercises 19-24, write the first five terms of a geometric sequence an based on the given information about the sequence. (See Example 2) a1=80andr=45 23PE 24PE 25PE For Exercises 25-30, write a formula for the nth term of the geometric sequence. (See Example 3) 3,6,12,24,For Exercises 25-30, write a formula for the nth term of the geometric sequence. (See Example 3) 2,1,12,14,28PE 29PE For Exercises 25-30, write a formula for the nth term of the geometric sequence. (See Example 3) 185,65,25,215,A farmer depreciates a 100,000 tractor. He estimates that the resale value of the tractor n years after purchase is 85 of its value from the previous year. a. Write a formula for the nth term of a sequence that represents the resale value of the tractor n years after purchase. b. What will the resale value be 5yr after purchase? Round to the nearest 1000 .A Coulter Counter is a device used to count the number of microscopic particles in a fluid, most notably, cells in blood. A hospital depreciates a 9000 Coulter Counter at a rate of 75 per year after purchase. a. Write a formula for the nth term of a sequence that represents the resale value of the device n years after purchase. b. What will the resale value be 4yr after purchase? Round to the nearest 100 .Doctors in a certain city report 24 confirmed cases of the flu to the health department. At that time, the health department declares a flu epidemic. If the number of reported cases increases by roughly 30 each week thereafter, find the number of cases 10 weeks after the initial report. Round to the nearest whole unit.After a 5-yr slump in the real estate market, housing prices stabilize and even begin to appreciate in value. One homeowner buys a house for 140,000 and finds that the value of the property increases by 3 per year thereafter. Assuming that the trend continues, find the value of the home 15yr later. Round to the nearest 1000 .35PE For Exercises 35-42, find the indicated term of a geometric sequence from the given information. (See Example 4) a1=16 and a2=12 . Find the fifth term.37PE 38PE 39PE For Exercises 35-42, find the indicated term of a geometric sequence from the given information. (See Example 4) a2=15 and r=13 . Find a8 .41PE 42PE If the second and third terms of a geometric sequence are 15 and 75 , what is the first term?44PE 45PE 46PE For Exercises 45-48, find a1 , and r for a geometric sequence an from the given information. (See Example 5) a3=72 and a6=2438 For Exercises 45-48, find a1 , and r for a geometric sequence an from the given information. (See Example 5) a3=45 and a6=24325 49PE 50PE 51PE For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) j=17234j1 For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) 15+5+53+59+527+581 54PE For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) 2+6+18++13,122 For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) 4+12+36++78,732 57PE 58PE For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) 1+15+125+1125+For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) 1+16+136+1216+For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) 21218132 62PE For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) 2+8+32+128+For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) 1+6+36+216+For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) j=123j1 For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) i=134i1 67PE 68PE For Exercises 49-72, find the sum of the geometric series, if possible. (See Examples 6-8) i=11242i 70PE 71PE 72PE 73PE For Exercises 73-80, write the repeating decimal as a fraction. (See Example 9) 0.2 75PE For Exercises 73-80, write the repeating decimal as a fraction. (See Example 9) 0.78 77PE For Exercises 73-80, write the repeating decimal as a fraction. (See Example 9) 0.72 79PE 80PE Bike Week in Daytona Beach brings an estimated 500,000 people to the town. Suppose that each person spends an average of 300 . a. How much money is infused into the local economy during Bike Week? b. If the money is respent in the community over and over again at a rate of 68 , determine the total amount spent. Assume that the money is respent an infinite number of times. (See Example 10)An individual with questionable integrity prints and spends 12,000 in counterfeit money. If the â€œmoneyâ€� is respent over and over again each time at a rate of 76 , determine the total amount spent. Assume that the â€œmoneyâ€� is respent an infinite number of times without being detected.Rafael received an inheritance of 18,000 . He saves 6480 and then spends 11,520 of the money on college tuition, books, and living expenses for school. If the money is respent over and over again in the community an infinite number of times, at a rate of 64 , determine the total amount spent.A tax rebate returns 100 million to individuals in the community. Suppose that 25,000,000 is put into savings, and that 75,000,000 is spent. If the money is spent over and over again an infinite number of times, each time at a rate of 75 , determine the total amount spent.85PE 86PE 87PE a. An employee invests 500 per month in an ordinary annuity. If the interest rate is 5 , find the value of the annuity after 18yr . b. If the employee invests 1000 per month in the annuity instead of 500 at 5 interest, find the value of the annuity after 18yr . Compare the result to part (a), c. If the employee invests $500 per month in the annuity at 5 interest find the value of the annuity after 36yr . Compare the result to part (a).89PE 90PE 91PE 92PE 93PE 94PE 95PE The initial swing (one way) of a pendulum makes an arc of 4ft . Each swing (one way) thereafter makes an arc of 90 of the length of the previous swing. What is the total arc length that the pendulum travels?97PE 98PE Suppose that an individual is paid 0.01 on day 1 and every day thereafter, the payment is doubled. a. Write a formula for the nth term of a sequence that gives the payment (in ) on day n . b. How much will the individual earn on day 10 ? Day 20 ? And day 30 ? c. What is the total amount earned in 30 days?The vibration of sound is measured in cycles per second, also called hertz (Hz). The frequency for middle C on a piano is 256Hz . The C above middle C (one octave above) is 512Hz . The frequencies of musical notes follow a geometric progression. a. Find the frequency for C two octaves above middle C . b. Find the frequency for C one octave below middle C .The yearly salary for job A is $60,000 initially with an annual raise of 3000 every year thereafter. The yearly salary for job B is 556,000 for year 1 with an annual raise of 6 . a. Consider a sequence representing the salary for job A for year n . Is this an arithmetic or geometric sequence? Find the total earnings for job A over 20yr . b. Consider a sequence representing the salary for job B for year n . Is this an arithmetic or geometric sequence? Find the total earnings for job B over 20yr . Round to the nearest dollar, c. What is the difference in total salary between the two jobs over 20yr ?102PE If a fair coin is flipped n times, the number of head/tail arrangements follows a geometric sequence. In the figure, if the coin is flipped 1 time, there are two possible outcomes, H or T . If the coin is flipped 2 times, then there are four possible outcomes: HH,HT,TH , and TT . a. Write a formula for the nth term of a sequence representing the number of outcomes if a fair coin is flipped n times. b. How many outcomes are there if a fair coin is flipped 10 times?104PE 105PE Explain why a finite number of terms is not sufficient to determine whether an infinite sequence is arithmetic or geometric. For example, explain why 4,16, can be arithmetic, geometric, or neither.107PE Show that x,x+2,x+4,x+6, is not a geometric sequence.109PE If a1,a2,a3, is an geometric sequence with common difference d . Show that 10a1,10a2,10a3,10a4, is a geometric sequence and find the common ratio r .Suppose that a1,a2,a3, is an arithmetic sequence with r0 and a10 , show that the sequence loga1,loga2,loga3, is arithmetic and find the common difference d .112PE Determine whether the sequence ln1,ln2,ln4,ln8, is arithmetic or geometric, if the sequence is arithmetic, find d . If the sequence is geometric, find r .For Exercises 1-10, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d. If the sequence is geometric, find the common ratio r. an=2n3+14 For Exercises 1-10, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d. If the sequence is geometric, find the common ratio r. an=35n13 For Exercises 1-10, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d. If the sequence is geometric, find the common ratio r. an=1n1 4PRE For Exercises 1-10, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d . If the sequence is geometric, find the common ratio r . an=n1n+1 For Exercises 1-10, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d . If the sequence is geometric, find the common ratio r . an=3n42n+1 For Exercises 1-10, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d . If the sequence is geometric, find the common ratio r . an=55n1 8PRE 9PRE 10PRE 11PRE 12PRE For Exercises 11-28, evaluate the sum if possible. i=17161i+1 14PRE 15PRE 16PRE 17PRE For Exercises 11-28, evaluate the sum if possible. n=15234n1 For Exercises 11-28, evaluate the sum if possible. n=1653n1 20PRE 21PRE 22PRE 23PRE 24PRE For Exercises 11-28, evaluate the sum if possible. 36+30+25+1256+26PRE 27PRE For Exercises 11-28, evaluate the sum if possible. 36+29+22+15++419 Use mathematical induction to prove Pn;2+4+6+8++2n=nn+1 2SP 3SP 4SP Let Pn be a statement involving the positive integer n and let k be an arbitrary positive integer. Proof by mathematical indicates that Pn is true for all positive integers n if (1) is true, and (2) the truth of Pk implies the truth of .The statement that Pk is true is called the hypothesis.3PE For Exercises 3-16, use mathematical induction to prove the given statement for all positive integers n . (See Examples 1-2) 2+8+14++6n4=n3n1 5PE 6PE 7PE 8PE 9PE 10PE 11PE For Exercises 3-16, use mathematical induction to prove the given statement for all positive integers n . (See Examples 1-2) 12+14+18++12n=112n 13PE 14PE 15PE 16PE For Exercises 17-24, use mathematical induction to prove the given statement for all positive integers n . (See Example 3) i=1n1=n 18PE 19PE 20PE 21PE 22PE For Exercises 17-24, use mathematical induction to prove the given statement for all positive integers n . (See Example 3) 4n1 is divisible by 3 .24PE 25PE 26PE 27PE 28PE 29PE 30PE For Exercises 29-32, use mathematical induction to prove the given statement. (See Example 4) 3n2n for positive integers n4 .32PE For Exercises 33-36, use mathematical induction to prove the given statement for all positive integers n and real numbers x and y . xyn=xnyn For Exercises 33-36, use mathematical induction to prove the given statement for all positive integers n and real numbers x and y . xyn=xnyn provided that y0 .For Exercises 33-36, use mathematical induction to prove the given statement for all positive integers n and real numbers x and y . If x1 , then xnxn1 .For Exercises 33-36, use mathematical induction to prove the given statement for all positive integers n and real numbers x and y . If 0x1 , then xnxn1 .37PE 38PE 39PE 40PE 41PE 42PE Exercises 43-44, use the Fibonacci sequence Fn=1,1,2,3,5,8,13, . Recall that the Fibonacci sequence can be defined recursively as F1=1,F2=1 , and Fn=Fn1+Fn2 for n3 . Prove that F1+F2+F3++Fn=Fn+21 for positive integers n3 .Exercises 43-44, use the Fibonacci sequence Fn=1,1,2,3,5,8,13, . Recall that the Fibonacci sequence can be defined recursively as F1=1,F2=1 , and Fn=Fn1+Fn2 for n3 . Prove that F1+F3+F5++F2n1=F2n for all positive integers n .Expand. a+b7 2SP Expand by using the binomial theorem. 4y+54 4SP Find the seventh term of 3c+d59 .The expression a3+3a2b+3ab2+b3 is called the expansion of a+b3 .2PE Consider a+bn , where n is a whole number. What is the degree of each term in the expansion?Consider a+bn , where n is a whole number. The coefficients of the terms in the expansion can be found by using triangle of by using nr .For positive integers n and kkn+ , the kth term of a+bn is given by ab .Given a+b17 , the 12th term is given by ab .7PE a. Write the first four rows of Pascalâ€™s triangle. b. Write the expansion of cd3 .9PE For Exercises 9-10, evaluate the given expressions. Compare the results to the fifth and sixth rows of Pascal's triangle. (See Example 2) a. 50 b. 51 c. 52 d. 53 e. 54 f. 55 For Exercises 11-14, evaluate the expression. 133 For Exercises 11-14, evaluate the expression. 1715 For Exercises 11-14, evaluate the expression. 115 14PE For Exercises 15-28, expand the binomial by using the binomial theorem. (See Examples 3-4) 3x+15 16PE For Exercises 15-28, expand the binomial by using the binomial theorem. (See Examples 3-4) 7x+33 18PE 19PE For Exercises 15-28, expand the binomial by using the binomial theorem. (See Examples 3-4) 4x14 21PE 22PE For Exercises 15-28, expand the binomial by using the binomial theorem. (See Examples 3-4) p2w46 24PE For Exercises 15-28, expand the binomial by using the binomial theorem. (See Examples 3-4) 0.2+0.1k4 26PE 27PE 28PE 29PE For Exorcises 29-40, find the indicated term of the binomial expansion. (See Example 5) p+q11 ; ninth term For Exorcises 29-40, find the indicated term of the binomial expansion. (See Example 5) cd8 ; fourth term 32PE For Exorcises 29-40, find the indicated term of the binomial expansion. (See Example 5) u2+2v415 ; tenth term For Exorcises 29-40, find the indicated term of the binomial expansion. (See Example 5) y3+2z214 ; tenth term For Exorcises 29-40, find the indicated term of the binomial expansion. (See Example 5) 3x2+y39 ; fourth term 36PE 37PE 38PE 39PE 40PE Expand exex4 .42PE 43PE 44PE 45PE 46PE Simplify. x+y3xy3 48PE 49PE 50PE For Exercises 49-52, simplify the difference quotient: fx+hfxh fx=x45x2+1 For Exercises 49-52, simplify the difference quotient: fx+hfxh fx=x46x24 For Exercises 53-56, use the binomial theorem to find the value of the complex number raised to the given power. Recall that i=1 . 2+i3

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