 12.1: write the rst four terms of the sequence. Assume n starts at 1.
 12.2: write the rst four terms of the sequence. Assume n starts at 1.
 12.3: write the rst four terms of the sequence. Assume n starts at 1.
 12.4: write the rst four terms of the sequence. Assume n starts at 1.
 12.5: write the rst four terms of the sequence. Assume n starts at 1.
 12.6: write the rst four terms of the sequence. Assume n starts at 1.
 12.7: write the rst four terms of the sequence. Assume n starts at 1.
 12.8: write the rst four terms of the sequence. Assume n starts at 1.
 12.9: write the rst four terms of the sequence. Assume n starts at 1.
 12.10: write the rst four terms of the sequence. Assume n starts at 1.
 12.11: write the rst four terms of the sequence. Assume n starts at 1.
 12.12: write the rst four terms of the sequence. Assume n starts at 1.
 12.13: nd the indicated term of the sequence.
 12.14: nd the indicated term of the sequence.
 12.15: nd the indicated term of the sequence.
 12.16: nd the indicated term of the sequence.
 12.17: nd the indicated term of the sequence.
 12.18: nd the indicated term of the sequence.
 12.19: nd the indicated term of the sequence.
 12.20: nd the indicated term of the sequence.
 12.21: write an expression for the nth term of the given sequence.2, 4, 6,...
 12.22: write an expression for the nth term of the given sequence. 3, 6, 9...
 12.23: write an expression for the nth term of the given sequence.
 12.24: write an expression for the nth term of the given sequence.1 2 , 1 ...
 12.25: write an expression for the nth term of the given sequence.
 12.26: write an expression for the nth term of the given sequence.
 12.27: write an expression for the nth term of the given sequence.
 12.28: write an expression for the nth term of the given sequence.1 3 , 2 ...
 12.29: simplify the ratio of factorials.
 12.30: simplify the ratio of factorials.
 12.31: simplify the ratio of factorials.
 12.32: simplify the ratio of factorials.
 12.33: simplify the ratio of factorials.
 12.34: simplify the ratio of factorials.
 12.35: simplify the ratio of factorials.
 12.36: simplify the ratio of factorials.
 12.37: simplify the ratio of factorials.
 12.38: simplify the ratio of factorials.
 12.39: simplify the ratio of factorials.
 12.40: simplify the ratio of factorials.
 12.41: write the rst four terms of the sequence dened by the recursion for...
 12.42: write the rst four terms of the sequence dened by the recursion for...
 12.43: write the rst four terms of the sequence dened by the recursion for...
 12.44: write the rst four terms of the sequence dened by the recursion for...
 12.45: write the rst four terms of the sequence dened by the recursion for...
 12.46: write the rst four terms of the sequence dened by the recursion for...
 12.47: write the rst four terms of the sequence dened by the recursion for...
 12.48: write the rst four terms of the sequence dened by the recursion for...
 12.49: write the rst four terms of the sequence dened by the recursion for...
 12.50: write the rst four terms of the sequence dened by the recursion for...
 12.51: evaluate the nite series.
 12.52: evaluate the nite series.
 12.53: evaluate the nite series.
 12.54: evaluate the nite series.
 12.55: evaluate the nite series.
 12.56: evaluate the nite series.
 12.57: evaluate the nite series.
 12.58: evaluate the nite series.
 12.59: evaluate the nite series.
 12.60: evaluate the nite series.
 12.61: evaluate the nite series.
 12.62: evaluate the nite series.
 12.63: evaluate the nite series.
 12.64: evaluate the nite series.
 12.65: , evaluate the innite series, if possible.
 12.66: , evaluate the innite series, if possible.
 12.67: , evaluate the innite series, if possible.
 12.68: , evaluate the innite series, if possible.
 12.69: , apply sigma notation to write the sum.
 12.70: , apply sigma notation to write the sum.
 12.71: , apply sigma notation to write the sum.
 12.72: , apply sigma notation to write the sum.1 + 2 + 3 + 4 + 5 + + 21 + ...
 12.73: , apply sigma notation to write the sum.
 12.74: , apply sigma notation to write the sum.
 12.75: , apply sigma notation to write the sum.
 12.76: apply sigma notation to write the sum.x + x2 + x3 2 + x4 6 + x5 24 ...
 12.77: Money. Upon graduation Jessica receives a commission from the U.S. ...
 12.78: Money. Dylan sells his car in his freshman year and puts $7000 in a...
 12.79: Salary.An attorney is trying to calculate the costs associated with...
 12.80: NFL Salaries.A player in the NFL typically has a career that lasts ...
 12.81: Salary. Upon graduation Sheldon decides to go to work for a local p...
 12.82: Escherichia coli.A single cell of bacteria reproduces through a pro...
 12.83: AIDS/HIV.A typical person has 500 to 1500 T cells per drop of blood...
 12.84: Company Sales. Lowes reported total sales from 2003 through 2004 in...
 12.85: Cost of Eating Out.A college student tries to save money by bringin...
 12.86: Cost of Acrylic Nails.A college student tries to save money by grow...
 12.87: Math and Engineering. The formula can be used to approximate the fu...
 12.88: Home Prices. If the ination rate is per year and the average price ...
 12.89: Approximating Functions. Polynomials can be used to approximate tra...
 12.90: Future Value of an Annuity.The future value of an ordinary annuity ...
 12.91: explain the mistake that is made.Simplify the ratio of factorials: ...
 12.92: explain the mistake that is made.Simplify the factorial expression:...
 12.93: explain the mistake that is made. Find the rst four terms of the se...
 12.94: explain the mistake that is made.Evaluate the series Solution: Writ...
 12.95: determine whether each statement is true or false.
 12.96: determine whether each statement is true or false.
 12.97: determine whether each statement is true or false.
 12.98: determine whether each statement is true or false.
 12.99: Write the rst four terms of the sequence dened by the recursion for...
 12.100: Write the rst four terms of the sequence dened by the recursion for...
 12.101: Fibonacci Sequence. An explicit formula for the nth term of the Fib...
 12.102: Let for and Find the rst ve terms of this sequence and make a gener...
 12.103: The sequence dened by approaches the number e as n gets large. Use ...
 12.104: The Fibonacci sequence is dened by and for The ratio is an an+1 an ...
 12.105:
 12.106: Use a graphing calculator SUMto sum . Compare it with your answer t...
 12.107: Use a graphing calculator SUM to sum . Compare it with your answer ...
 12.1: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.2: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.3: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.4: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.5: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.6: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.7: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.8: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.9: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.10: determine whether the sequence is arithmetic. If it is, nd the comm...
 12.11: nd the rst four terms of the sequence. Determine whether the sequen...
 12.12: nd the rst four terms of the sequence. Determine whether the sequen...
 12.13: nd the rst four terms of the sequence. Determine whether the sequen...
 12.14: nd the rst four terms of the sequence. Determine whether the sequen...
 12.15: nd the rst four terms of the sequence. Determine whether the sequen...
 12.16: nd the rst four terms of the sequence. Determine whether the sequen...
 12.17: nd the rst four terms of the sequence. Determine whether the sequen...
 12.18: nd the rst four terms of the sequence. Determine whether the sequen...
 12.19: nd the rst four terms of the sequence. Determine whether the sequen...
 12.20: nd the rst four terms of the sequence. Determine whether the sequen...
 12.21: nd the specied term for each arithmetic sequence.
 12.22: nd the specied term for each arithmetic sequence.
 12.23: nd the specied term for each arithmetic sequence.
 12.24: nd the specied term for each arithmetic sequence.
 12.25: nd the specied term for each arithmetic sequence.
 12.26: nd the specied term for each arithmetic sequence.
 12.27: nd the specied term for each arithmetic sequence.
 12.28: nd the specied term for each arithmetic sequence.
 12.29: nd the specied term for each arithmetic sequence The 10th term of t...
 12.30: nd the specied term for each arithmetic sequence 31. The 100th term...
 12.31: nd the specied term for each arithmetic sequenceThe 100th term of t...
 12.32: nd the specied term for each arithmetic sequence
 12.33: nd the specied term for each arithmetic sequence
 12.34: nd the specied term for each arithmetic sequence The 33rd term of t...
 12.35: for each arithmetic sequence described, nd a1 and d and construct t...
 12.36: for each arithmetic sequence described, nd a1 and d and construct t...
 12.37: for each arithmetic sequence described, nd a1 and d and construct t...
 12.38: for each arithmetic sequence described, nd a1 and d and construct t...
 12.39: for each arithmetic sequence described, nd a1 and d and construct t...
 12.40: for each arithmetic sequence described, nd a1 and d and construct t...
 12.41: nd the sum.
 12.42: nd the sum.
 12.43: nd the sum.
 12.44: nd the sum.
 12.45: nd the sum.
 12.46: nd the sum.
 12.47: nd the sum.2 + 7 + 12 + 17 + + 62
 12.48: nd the sum.
 12.49: nd the sum.4 + 7 + 10 + + 151
 12.50: nd the sum.
 12.51: nd the sum.
 12.52: nd the sum.
 12.53: nd the partial sum of the arithmetic series. The rst 18 terms of .
 12.54: nd the partial sum of the arithmetic series. The rst 21 terms of .
 12.55: nd the partial sum of the arithmetic series. The rst 43 terms of
 12.56: nd the partial sum of the arithmetic series. The rst 37 terms of
 12.57: nd the partial sum of the arithmetic series. The rst 18 terms of .
 12.58: nd the partial sum of the arithmetic series.
 12.59: Comparing Salaries. Colin and Camden are twin brothers graduating w...
 12.60: Comparing Salaries. On graduating with a Ph.D. in optical sciences,...
 12.61: Theater Seating.You walk into the premiere of Brad Pitts new movie,...
 12.62: Field of Tulips. Every spring the Skagit County Tulip Festival plan...
 12.63: Worlds Largest Champagne Fountain. From December 28 to 30, 1999, Lu...
 12.64: Stacking of Logs. If 25 logs are laid side by side on the ground, a...
 12.65: Falling Object.When a skydiver jumps out of an airplane, she falls ...
 12.66: Falling Object. If a penny is dropped out of a plane, it falls appr...
 12.67: Grocery Store.A grocer has a triangular display of oranges in a win...
 12.68: Salary. Suppose your salary is $45,000 and you receive a $1500 rais...
 12.69: Theater Seating.At a theater, seats are arranged in a triangular pa...
 12.70: Mathematics. Find the exact sum of 1 e +3 e +5 e + + 23 e
 12.71: Find the general or nth term of the arithmetic sequence 3, 4, 5, 6,...
 12.72: Find the general or nth term of the arithmetic sequence 10, 8, 6, ....
 12.73: Find the sum Solution: The sum is given by where Identify the 1st a...
 12.74: Find the sum . Solution: This is an arithmetic sequence with common...
 12.75: determine whether each statement is true or false. An arithmetic se...
 12.76: determine whether each statement is true or false. The sum of all i...
 12.77: An alternating sequence cannot be an arithmetic sequence.
 12.78: The common difference of an arithmetic sequence is always positive.
 12.79: Find the sum
 12.80: Find the sum a30 k=29 ln ek.
 12.81: The wave number, v (reciprocal of wave length) of certain light wav...
 12.82: In a certain arithmetic sequence and If nd the value of n.
 12.83: Use a graphing calculator SUM to sum the natural numbers from 1 to ...
 12.84: Use a graphing calculator to sum the even natural numbers from 1 to...
 12.85: Use a graphing calculator to sum the odd natural numbers from 1 to ...
 12.86: Use a graphing calculator to sum Compare it with your answer to Exe...
 12.87: Use a graphing calculator to sum
 12.88: Use a graphing calculator to sum
 12.1: determine whether the sequence is geometric. If it is, nd the commo...
 12.36: determine whether the sequence is geometric. If it is, nd the commo...
 12.37: determine whether the sequence is geometric. If it is, nd the commo...
 12.38: determine whether the sequence is geometric. If it is, nd the commo...
 12.39: determine whether the sequence is geometric. If it is, nd the commo...
 12.40: determine whether the sequence is geometric. If it is, nd the commo...
 12.41: determine whether the sequence is geometric. If it is, nd the commo...
 12.42: determine whether the sequence is geometric. If it is, nd the commo...
 12.43: write the rst ve terms of the geometric series.
 12.44: write the rst ve terms of the geometric series.
 12.45: write the rst ve terms of the geometric series.
 12.46: write the rst ve terms of the geometric series.
 12.47: write the rst ve terms of the geometric series.
 12.48: write the rst ve terms of the geometric series.
 12.49: write the rst ve terms of the geometric series.
 12.50: write the rst ve terms of the geometric series.
 12.51: write the formula for the nth term of the geometric series.
 12.52: write the formula for the nth term of the geometric series.
 12.53: write the formula for the nth term of the geometric series.
 12.54: write the formula for the nth term of the geometric series.
 12.55: write the formula for the nth term of the geometric series.
 12.56: write the formula for the nth term of the geometric series.
 12.57: write the formula for the nth term of the geometric series.
 12.58: write the formula for the nth term of the geometric series.
 12.59: nd the indicated term of the geometric sequence. 7th term of the se...
 12.60: nd the indicated term of the geometric sequence. 10th term of the s...
 12.61: nd the indicated term of the geometric sequence. 13th term of the s...
 12.62: nd the indicated term of the geometric sequence. 9th term of the se...
 12.63: nd the indicated term of the geometric sequence. 15th term of the s...
 12.64: nd the indicated term of the geometric sequence. 8th term of the se...
 12.65: nd the sum of the nite geometric series.
 12.66: nd the sum of the nite geometric series.
 12.67: nd the sum of the nite geometric series.
 12.68: nd the sum of the nite geometric series.
 12.69: nd the sum of the nite geometric series.
 12.70: nd the sum of the nite geometric series.
 12.71: nd the sum of the nite geometric series.
 12.72: nd the sum of the nite geometric series.
 12.73: nd the sum of the nite geometric series.
 12.74: nd the sum of the nite geometric series.
 12.75: nd the sum of the innite geometric series, if possible.
 12.76: nd the sum of the innite geometric series, if possible.
 12.77: nd the sum of the innite geometric series, if possible.
 12.78: nd the sum of the innite geometric series, if possible.
 12.79: nd the sum of the innite geometric series, if possible.
 12.80: nd the sum of the innite geometric series, if possible.
 12.81: nd the sum of the innite geometric series, if possible.
 12.82: nd the sum of the innite geometric series, if possible.
 12.83: nd the sum of the innite geometric series, if possible.
 12.84: nd the sum of the innite geometric series, if possible.
 12.85: nd the sum of the innite geometric series, if possible.
 12.86: nd the sum of the innite geometric series, if possible.
 12.87: nd the sum of the innite geometric series, if possible.
 12.88: nd the sum of the innite geometric series, if possible.
 12.89: Salary. Jeremy is offered a government job with the Department of C...
 12.90: Salary.Alison is offered a job with a small startup company that w...
 12.91: Depreciation. Brittany, a graduating senior in high school, receive...
 12.92: Depreciation. Derek is deciding between a new Honda Accord and the ...
 12.93: Bungee Jumping.A bungee jumper rebounds 70% of the height jumped. A...
 12.94: Bungee Jumping.A bungee jumper rebounds 65% of the height jumped. A...
 12.95: Population Growth. One of the fastestgrowing universities in the c...
 12.96: Website Hits. The website for Matchbox 20 (www. matchboxtwenty.com)...
 12.97: Rich Mans Promise.A rich man promises that he will give you $1000 o...
 12.98: Poor Mans Clever Deal.A poor man promises to work for you for $0.01...
 12.99: Investing Lunch.A newlywed couple decides to stop going out to lunc...
 12.66: Pizza as an Investment.A college freshman decides to stop ordering ...
 12.67: TaxDeferred Annuity. Dr. Schober contributes $500 from her paychec...
 12.68: Saving for a House. If a new graduate decides she wants to save for...
 12.69: House Values. In 2008, you buy a house for $195,000. The value of t...
 12.70: The Bouncing Ball Problem.A ball is dropped from a height of 9 feet...
 12.71: Probability. A fair coin is tossed repeatedly. The probability that...
 12.72: Salary. Suppose you work for a supervisor who gives you two differe...
 12.73: explain the mistake that is made.nd the nth term of the geometric s...
 12.74: explain the mistake that is made.Find the sum of the rst n terms of...
 12.75: explain the mistake that is made.Find the sum of the nite geometric...
 12.76: explain the mistake that is made.Find the sum of the innite geometr...
 12.77: determine whether each statement is true or false. An alternating s...
 12.78: determine whether each statement is true or false. All nite and inn...
 12.79: determine whether each statement is true or false. The common ratio...
 12.80: determine whether each statement is true or false. An innite geomet...
 12.81: State the conditions for the sumto exist. Assuming those conditions...
 12.82: Find the sum of
 12.83: Represent the repeating decimal 0.474747. . . as a fraction (ratio ...
 12.84: Suppose the sum of an innite geometric series is where x is a varia...
 12.85: Sum the series: Apply a graphing utility to conrm your answer.
 12.86: Does the sum of the innite series exist? Use a graphing calculator ...
 12.87: Apply a graphing utility to plot and and let the range of x be Base...
 12.88: Apply a graphing utility to plot and and let x range from Based on ...
 12.89: Apply a graphing utility to plotand and let x range from Based on w...
 12.1: prove the statements using mathematical induction for all positive ...
 12.2: prove the statements using mathematical induction for all positive ...
 12.3: prove the statements using mathematical induction for all positive ...
 12.4: prove the statements using mathematical induction for all positive ...
 12.5: prove the statements using mathematical induction for all positive ...
 12.6: prove the statements using mathematical induction for all positive ...
 12.7: prove the statements using mathematical induction for all positive ...
 12.8: prove the statements using mathematical induction for all positive ...
 12.9: prove the statements using mathematical induction for all positive ...
 12.10: prove the statements using mathematical induction for all positive ...
 12.11: prove the statements using mathematical induction for all positive ...
 12.12: prove the statements using mathematical induction for all positive ...
 12.13: prove the statements using mathematical induction for all positive ...
 12.14: prove the statements using mathematical induction for all positive ...
 12.15: prove the statements using mathematical induction for all positive ...
 12.16: prove the statements using mathematical induction for all positive ...
 12.17: prove the statements using mathematical induction for all positive ...
 12.18: prove the statements using mathematical induction for all positive ...
 12.19: prove the statements using mathematical induction for all positive ...
 12.20: prove the statements using mathematical induction for all positive ...
 12.21: prove the statements using mathematical induction for all positive ...
 12.22: prove the statements using mathematical induction for all positive ...
 12.23: prove the statements using mathematical induction for all positive ...
 12.24: prove the statements using mathematical induction for all positive ...
 12.25: The Tower of Hanoi. This is a game with three pegs and n disks (lar...
 12.26: The Tower of Hanoi. This is a game with three pegs and n disks (lar...
 12.27: The Tower of Hanoi. This is a game with three pegs and n disks (lar...
 12.28: The Tower of Hanoi. This is a game with three pegs and n disks (lar...
 12.29: Telephone Infrastructure. Suppose there are n cities that are to be...
 12.30: The Tower of Hanoi. This is a game with three pegs and n disks (lar...
 12.31: determine whether each statement is true or false. Assume is true. ...
 12.32: determine whether each statement is true or false. Assume is true. ...
 12.33: Apply mathematical induction to prove:
 12.34: Apply mathematical induction to prove:a n k=1k5 =n2(n + 1)2A2n2 + 2...
 12.35: Apply mathematical induction to prove:
 12.36: Use a graphing calculator to sum the series (1 2) (2 3) (3 4) . . ....
 12.37: evaluate the expression on the right side for . Do they agree with ...
 12.38: Use a graphing calculator to sum the series . . . on the left side,...
 12.1: evaluate the binomial coefcients.
 12.2: evaluate the binomial coefcients.
 12.3: evaluate the binomial coefcients.
 12.4: evaluate the binomial coefcients.
 12.5: evaluate the binomial coefcients.
 12.6: evaluate the binomial coefcients.
 12.7: evaluate the binomial coefcients.
 12.8: evaluate the binomial coefcients.
 12.9: evaluate the binomial coefcients.
 12.10: evaluate the binomial coefcients.
 12.11: expand the expression using the binomial theorem.
 12.12: expand the expression using the binomial theorem.
 12.13: expand the expression using the binomial theorem.
 12.14: expand the expression using the binomial theorem.
 12.15: expand the expression using the binomial theorem.
 12.16: expand the expression using the binomial theorem.
 12.17: expand the expression using the binomial theorem.
 12.18: expand the expression using the binomial theorem.
 12.19: expand the expression using the binomial theorem.
 12.20: expand the expression using the binomial theorem.
 12.21: expand the expression using the binomial theorem.
 12.22: expand the expression using the binomial theorem.
 12.23: expand the expression using the binomial theorem.
 12.24: expand the expression using the binomial theorem.
 12.25: expand the expression using the binomial theorem.
 12.26: expand the expression using the binomial theorem.
 12.27: expand the expression using the binomial theorem.
 12.28: expand the expression using the binomial theorem.
 12.29: expand the expression using the binomial theorem.
 12.30: expand the expression using the binomial theorem.
 12.31: expand the expression using the binomial theorem.
 12.32: expand the expression using the binomial theorem.
 12.33: expand the expression using Pascals triangle.
 12.34: expand the expression using Pascals triangle.
 12.35: expand the expression using Pascals triangle.
 12.36: expand the expression using Pascals triangle.
 12.37: nd the coefcient C of the term in the binomial expansion.
 12.38: nd the coefcient C of the term in the binomial expansion.
 12.39: nd the coefcient C of the term in the binomial expansion.
 12.40: nd the coefcient C of the term in the binomial expansion.
 12.41: nd the coefcient C of the term in the binomial expansion.
 12.42: nd the coefcient C of the term in the binomial expansion.
 12.43: nd the coefcient C of the term in the binomial expansion.
 12.44: nd the coefcient C of the term in the binomial expansion.
 12.45: you will learn the n choose k notation for combinations. Lottery. I...
 12.46: you will learn the n choose k notation for combinations. Lottery. I...
 12.47: you will learn the n choose k notation for combinations. Poker.With...
 12.48: you will learn the n choose k notation for combinations. Canasta. I...
 12.49: explain the mistake that is made. Evaluate the expression Solution:...
 12.50: explain the mistake that is made. Expand Solution: Write out with b...
 12.51: determine whether each statement is true or false. The binomial exp...
 12.52: determine whether each statement is true or false. The binomial exp...
 12.53: determine whether each statement is true or false.
 12.54: determine whether each statement is true or false.
 12.55: Show that if 0 k n.
 12.56: Show that if n is a positive integer, then:Hint: Let and use the bi...
 12.57: With a graphing utility, plot and in the same viewing screen. What ...
 12.58: With a graphing utility, plot andWhat is the binomialexpansion of
 12.59: With a graphing utility, plot and for What do you notice happening ...
 12.60: With a graphing utility, plot and for What do you notice happening ...
 12.61: With a graphing utility, plot and for What do you notice happening ...
 12.62: With a graphing utility, plotand for What do you notice happening e...
 12.1: use the formula for to evaluate each expression.
 12.2: use the formula for to evaluate each expression.
 12.3: use the formula for to evaluate each expression.
 12.4: use the formula for to evaluate each expression.
 12.5: use the formula for to evaluate each expression.
 12.6: use the formula for to evaluate each expression.
 12.7: use the formula for to evaluate each expression.
 12.8: use the formula for to evaluate each expression.
 12.9: use the formula for to evaluate each expression.10C5
 12.10: use the formula for to evaluate each expression.9C4
 12.11: use the formula for to evaluate each expression.50C6
 12.12: use the formula for to evaluate each expression.50C10
 12.13: use the formula for to evaluate each expression.
 12.14: use the formula for to evaluate each expression.. 8C8
 12.15: use the formula for to evaluate each expression.30C4
 12.16: use the formula for to evaluate each expression.13C5
 12.17: use the formula for to evaluate each expression.45C8
 12.18: use the formula for to evaluate each expression.30C4
 12.19: Computers.At the www.dell.com website, a customer can build a syste...
 12.20: Houses. In a new home community, a person can select from one of fo...
 12.21: Wedding Invitations.An engaged couple is ordering wedding invitatio...
 12.22: Dinner. Siblings are planning their fathers 65th birthday dinner an...
 12.23: PIN Number. Most banks require a 4digit ATM PIN code for each cust...
 12.24: Password.All email accounts require passwords. If a fourcharacter...
 12.25: Leadership. There are 15 professors in a department and there are f...
 12.26: Fraternity Elections.A fraternity is having elections. There are th...
 12.27: MultipleChoice Tests. There are 20 questions on a multiplechoice ...
 12.28: MultipleChoice Tests. There are 25 questions on a multiplechoice ...
 12.29: Zip Codes. In the United States a 5digit zip code is used to route...
 12.30: License Plates. In a particular state there are six characters in a...
 12.31: Class Seating. If there are 30 students in a class and there are ex...
 12.32: Season Tickets. Four friends buy four season tickets to the Green B...
 12.33: Combination Permutation Lock.A combination lock on most lockers wil...
 12.34: Safe.A safe will open when the correct choice of three numbers (1 t...
 12.35: Rafe.A fundraiser rafe is held to benet cystic brosis research, and...
 12.36: Ironman Triathlon. If 100 people compete in an ironman triathlon, h...
 12.37: Lotto. If a state lottery picks from 53 numbers and 6 numbers are s...
 12.38: Lotto. If a state lottery picks from 53 numbers and 5 numbers are s...
 12.39: Cards. In a deck of 52 cards, how many different 5card hands can b...
 12.40: Cards. In a deck of 52 cards, how many different 7card hands can b...
 12.41: Blackjack. In a singledeck blackjack game (52 cards), how many dif...
 12.42: Blackjack. In a single deck, how many twocard combinations are the...
 12.43: March Madness. Every spring, the NCAA mens basketball tournament st...
 12.44: March Madness. Every spring, the NCAA mens basketball tournament st...
 12.45: NFL Playoffs. There are 32 teams in the National Football League (1...
 12.46: NFL Playoffs.After the regular season in the National Football Leag...
 12.47: Survivor. On the television show Survivor, one person is voted off ...
 12.48: American Idol. On the television show American Idol,a young rising ...
 12.49: Dancing with the Stars. In the popular TV show Dancing with the Sta...
 12.50: Dancing with the Stars. See Exercise 49. How many ways can six male...
 12.51: explain the mistake that is made. "In a lottery that picks from 30 ...
 12.52: explain the mistake that is made. "A homeowners association has 12 ...
 12.53: determine whether each statement is true or false. The number of pe...
 12.54: determine whether each statement is true or false. The number of pe...
 12.55: determine whether each statement is true or false. The number of fo...
 12.56: determine whether each statement is true or false. The number of po...
 12.57: What is the relationship between and
 12.58: What is the relationship between and nPr1?
 12.59: Simplify the expression .
 12.60: What is the relationship between and nCnr?
 12.61: Employ a graphing utility with a feature and compare it with answer...
 12.62: Employ a graphing utility with a feature and compare it with answer...
 12.63: Use a graphing calculator to evaluate: a. b. c. Are answers in (a) ...
 12.64: Use a graphing calculator to evaluate: a. b. c. Are answers in (a) ...
 12.1: nd the sample space for each experiment. The sum of two dice rolled...
 12.2: nd the sample space for each experiment. A coin tossed three times ...
 12.3: nd the sample space for each experiment. The sex (boy or girl) of f...
 12.4: nd the sample space for each experiment. Tossing a coin and rolling...
 12.5: nd the sample space for each experiment. Two balls selected from a ...
 12.6: nd the sample space for each experiment. The grade (freshman, sopho...
 12.7: nd the probability for the experiment of tossing a coin three times...
 12.8: nd the probability for the experiment of tossing a coin three times...
 12.9: nd the probability for the experiment of tossing a coin three times...
 12.10: nd the probability for the experiment of tossing a coin three times...
 12.11: nd the probability for the experiment of tossing two dice. The sum ...
 12.12: nd the probability for the experiment of tossing two dice. The sum ...
 12.13: nd the probability for the experiment of tossing two dice. The sum ...
 12.14: nd the probability for the experiment of tossing two dice. The sum ...
 12.15: nd the probability for the experiment of tossing two dice. The sum ...
 12.16: nd the probability for the experiment of tossing two dice. The sum ...
 12.17: nd the probability for the experiment of drawing a single card from...
 12.18: nd the probability for the experiment of drawing a single card from...
 12.19: nd the probability for the experiment of drawing a single card from...
 12.20: nd the probability for the experiment of drawing a single card from...
 12.21: let and and nd the probability of the event. Probability of not occ...
 12.22: let and and nd the probability of the event. Probability of not occ...
 12.23: let and and nd the probability of the event. Probability of either ...
 12.24: let and and nd the probability of the event. Probability of either ...
 12.25: let and and nd the probability of the event. Probability of both an...
 12.26: let and and nd the probability of the event. Probability of both an...
 12.27: Cards.A deck of 52 cards is dealt. a. How many possible combination...
 12.28: Blackjack.A deck of 52 cards is dealt for blackjack. a. How many po...
 12.29: Cards.With a 52card deck,what is the probability of drawing a 7 or...
 12.30: Cards.With a 52card deck,what is the probability of drawing a red ...
 12.31: Cards. By drawing twice, what is the probability of drawing a 7 and...
 12.32: Cards. By drawing twice, what is the probability of drawing a red 7...
 12.33: Children.What is the probability of having ve daughters in a row an...
 12.34: Children. What is the probability of having four sons in a row and ...
 12.35: Children. What is the probability that of ve children at least one ...
 12.36: Children. What is the probability that of six children at least one...
 12.37: Roulette. In roulette, there are 38 numbered slots (136, 0, and 00)...
 12.38: Roulette. What is the probability of having 2 greens in a row on a ...
 12.39: Item Defectiveness. For a particular brand of DVD players, 10% of t...
 12.40: Item Defectiveness. For a particular brand of generators 20% of the...
 12.41: Number Generator. A randomnumber generator (computer program that ...
 12.42: Number Generator. A randomnumber generator is used to select two n...
 12.43: assume each deal is from a complete (shufed) single deck of cards. ...
 12.44: assume each deal is from a complete (shufed) single deck of cards. ...
 12.45: Sports. With the salary cap in the NFL, it is said that on any give...
 12.46: Sports. With the salary cap in the NFL, it is said that on any give...
 12.47: Genetics. Suppose both parents have the brown/blue pair of eyecolo...
 12.48: Genetics. Refer to Exercise 47. In this exercise, the father has a ...
 12.49: Playing Cards. How many 5card hands can be drawn from a 52card de...
 12.50: Playing Cards. a. How many ways can you select 2 aces and 3 other c...
 12.51: Playing Cards. Find the probability of drawing 5 clubs from a stand...
 12.52: Poker. Find the probability of getting 2 ves and 3 kings when drawi...
 12.53: explain the mistake that is made. Calculate the probability of draw...
 12.54: explain the mistake that is made. Calculate the probability of havi...
 12.55: determine whether each statement is true or false. If and then must...
 12.56: determine whether each statement is true or false.If two events are...
 12.57: determine whether each statement is true or false.If two events are...
 12.58: determine whether each statement is true or false.The probability o...
 12.59: If two people are selected at random, what is the probability that ...
 12.60: If 30 people are selected at random, what is the probability that a...
 12.61: If one die is weighted so that 3 and 4 are the only numbers that th...
 12.62: If one die is weighted so that 3 and 4 are the only numbers that th...
 12.63: Use a randomnumber generator on a graphing utility to select two n...
 12.64: Use a randomnumber generator on a graphing utility to select two n...
 12.65: when a die is rolled once,the probability of getting a 2 is and the...
 12.66: when a die is rolled once,the probability of getting a 2 is and the...
 12.1: Write the rst four terms of the sequence. Assume n starts at 1.
 12.2: Write the rst four terms of the sequence. Assume n starts at 1.
 12.3: Write the rst four terms of the sequence. Assume n starts at 1.
 12.4: Write the rst four terms of the sequence. Assume n starts at 1.
 12.5: Find the indicated term of the sequence.
 12.6: Find the indicated term of the sequence.
 12.7: Find the indicated term of the sequence.
 12.8: Find the indicated term of the sequence.
 12.9: Write an expression for the nth term of the given sequence.
 12.10: Write an expression for the nth term of the given sequence. 1, 1 2,...
 12.11: Write an expression for the nth term of the given sequence.
 12.12: Write an expression for the nth term of the given sequence. 1, 10, ...
 12.13: Simplify the ratio of factorials.
 12.14: Simplify the ratio of factorials.
 12.15: Simplify the ratio of factorials.
 12.16: Simplify the ratio of factorials.
 12.17: Write the rst four terms of the sequence dened by the recursion for...
 12.18: Write the rst four terms of the sequence dened by the recursion for...
 12.19: Write the rst four terms of the sequence dened by the recursion for...
 12.20: Write the rst four terms of the sequence dened by the recursion for...
 12.21: Evaluate the nite series.
 12.22: Evaluate the nite series.
 12.23: Evaluate the nite series.
 12.24: Evaluate the nite series.
 12.25: Use sigma (summation) notation to write the sum.
 12.26: Use sigma (summation) notation to write the sum. 2 + 4 + 6 + 8 + 10...
 12.27: Use sigma (summation) notation to write the sum. 1 + x + x2 2 + x3 ...
 12.28: Use sigma (summation) notation to write the sum. x  x2 + x3 2 x4 6...
 12.29: Marines Investment. With the prospect of continued ghting in Iraq, ...
 12.30: Sports. The NFL minimum salary for a rookie is $180,000. Suppose a ...
 12.31: Determine whether the sequence is arithmetic. If it is, nd the comm...
 12.32: Determine whether the sequence is arithmetic. If it is, nd the comm...
 12.33: Determine whether the sequence is arithmetic. If it is, nd the comm...
 12.34: Determine whether the sequence is arithmetic. If it is, nd the comm...
 12.35: Determine whether the sequence is arithmetic. If it is, nd the comm...
 12.36: Determine whether the sequence is arithmetic. If it is, nd the comm...
 12.37: Find the general, or nth, term of the arithmetic sequence given the...
 12.38: Find the general, or nth, term of the arithmetic sequence given the...
 12.39: Find the general, or nth, term of the arithmetic sequence given the...
 12.40: Find the general, or nth, term of the arithmetic sequence given the...
 12.41: For each arithmetic sequence described below, nd and d and construc...
 12.42: For each arithmetic sequence described below, nd and d and construc...
 12.43: For each arithmetic sequence described below, nd and d and construc...
 12.44: For each arithmetic sequence described below, nd and d and construc...
 12.45: Find the sum.
 12.46: Find the sum.
 12.47: Find the sum.2 + 8 + 14 + 20 + + 68
 12.48: Find the sum.1 4  1 4  3 4   31
 12.49: Salary. Upon graduating with M.B.A.s, Bob and Tania opt for differe...
 12.50: Gravity.When a skydiver jumps out of an airplane, she falls approxi...
 12.51: Determine whether the sequence is geometric. If it is, nd the commo...
 12.52: Determine whether the sequence is geometric. If it is, nd the commo...
 12.53: Determine whether the sequence is geometric. If it is, nd the commo...
 12.54: Determine whether the sequence is geometric. If it is, nd the commo...
 12.55: Write the rst ve terms of the geometric series.
 12.56: Write the rst ve terms of the geometric series.
 12.57: Write the rst ve terms of the geometric series.
 12.58: Write the rst ve terms of the geometric series.
 12.59: Write the formula for the nth term of the geometric series.
 12.60: Write the formula for the nth term of the geometric series.
 12.61: Write the formula for the nth term of the geometric series.
 12.62: Write the formula for the nth term of the geometric series.
 12.63: Find the indicated term of the geometric sequence. 25th term of the...
 12.64: Find the indicated term of the geometric sequence. 10th term of the...
 12.65: Find the indicated term of the geometric sequence. 12th term of the...
 12.66: Find the indicated term of the geometric sequence. 11th term of the...
 12.67: Evaluate the geometric series, if possible.
 12.68: Evaluate the geometric series, if possible.
 12.69: Evaluate the geometric series, if possible.
 12.70: Evaluate the geometric series, if possible.
 12.71: Evaluate the geometric series, if possible.
 12.72: Evaluate the geometric series, if possible.
 12.73: Salary. Murad is uent in four languages and is offered a job with t...
 12.74: Boat Depreciation. Upon graduating from Auburn University, Philip a...
 12.75: Prove the statements using mathematical induction for all positive ...
 12.76: Prove the statements using mathematical induction for all positive ...
 12.77: Prove the statements using mathematical induction for all positive ...
 12.78: Prove the statements using mathematical induction for all positive ...
 12.79: Evaluate the binomial coefcients.
 12.80: Evaluate the binomial coefcients.
 12.81: Evaluate the binomial coefcients.
 12.82: Evaluate the binomial coefcients.
 12.83: Expand the expression using the binomial theorem.
 12.84: Expand the expression using the binomial theorem.
 12.85: Expand the expression using the binomial theorem.
 12.86: Expand the expression using the binomial theorem.
 12.87: Expand the expression using the binomial theorem.
 12.88: Expand the expression using the binomial theorem.
 12.89: Expand the expression using Pascals triangle.
 12.90: Expand the expression using Pascals triangle.
 12.91: Find the coefcient C of the term in the binomial expansion.
 12.92: Find the coefcient C of the term in the binomial expansion.
 12.93: Find the coefcient C of the term in the binomial expansion.
 12.94: Find the coefcient C of the term in the binomial expansion.
 12.95: Lottery. In a state lottery in which 6 numbers are drawn from a pos...
 12.96: Canasta. In the card game canasta, two decks of cards including the...
 12.97: Use the formula for to evaluate each expression.
 12.98: Use the formula for to evaluate each expression.
 12.99: Use the formula for to evaluate each expression.
 12.100: Use the formula for to evaluate each expression.
 12.101: Use the formula for to evaluate each expression.
 12.102: Use the formula for to evaluate each expression.
 12.103: Use the formula for to evaluate each expression.
 12.104: Use the formula for to evaluate each expression.
 12.105: Car Options. A new Honda Accord comes in three models (LX,VX, and E...
 12.106: Email Passwords. All email accounts require passwords. If a sixc...
 12.107: Team Arrangements. There are 10 candidates for the board of directo...
 12.108: License Plates. In a particular state, there are six characters in ...
 12.109: Seating Arrangements. Five friends buy ve season tickets to the Phi...
 12.110: Safe. A safe will open when the correct choice of three numbers (1 ...
 12.111: Rafe. A fundraiser rafe is held to benet the Make a Wish Foundation...
 12.112: Sports. There are 117 Division 1A football teams in the United Sta...
 12.113: Cards. In a deck of 52 cards, how many different 6card hands can b...
 12.114: Blackjack. In a game of twodeck blackjack (104 cards), how many 2...
 12.115: Coin Tossing. For the experiment of tossing a coin four times, what...
 12.116: Dice. For an experiment of tossing two dice, what is the probabilit...
 12.117: Dice. For an experiment of tossing two dice, what is the probabilit...
 12.118: Cards. For a deck of 52 cards, what is the probability of drawing a...
 12.119: let and and nd the probability of the event. Probability. Find the ...
 12.120: let and and nd the probability of the event.Probability. Find the p...
 12.121: let and and nd the probability of the event.Probability. Find the p...
 12.122: let and and nd the probability of the event. Probability. Find the ...
 12.123: Cards. With a 52card deck, what is the probability of drawing an a...
 12.124: Cards. By drawing twice, what is the probability of drawing an ace ...
 12.125: Children. What is the probability that in a family of ve children a...
 12.126: Sports. With the salary cap in the NFL, it is said that on any give...
 12.127: Use a graphing calculator SUM to nd the sum of the series
 12.128: Use a graphing calculator SUM to nd the sum of the innite series if...
 12.129: Use a graphing calculator to sum
 12.130: Use a graphing calculator to sum
 12.131: Apply a graphing utility to plot and and let x range from . Based o...
 12.132: Does the sum of the innite series exist? Use a graphing calculator ...
 12.133: Use a graphing calculator to sum the series on the left side, and e...
 12.134: Use a graphing calculator to plot the graphs of and in the by viewi...
 12.135: With a graphing utility, plot y1 1 8x, y2 1 8x 24x2, y3 1 8x 24x2 3...
 12.136: With a graphing utility,plot and for What do you notice happening e...
 12.137: Use a graphing calculator to evaluate: a. b. 16C7 c. Are answers in...
 12.138: Use a graphing calculator to evaluate: a. b. 52C6 c. Are answers in...
 12.139: when two dice are rolled, the probability of getting a sum of 9 is ...
 12.140: when two dice are rolled, the probability of getting a sum of 9 is ...
 12.1: use the sequence 1, x, x2, x3, Write the nth term of the sequence.
 12.2: use the sequence 1, x, x2, x3, Classify this sequence as arithmetic...
 12.3: use the sequence 1, x, x2, x3, Find the nth partial sum of the series
 12.4: use the sequence 1, x, x2, x3, Assuming this sequence is innite, wr...
 12.5: use the sequence 1, x, x2, x3, Assuming this sequence is innite, wh...
 12.6: Find the following sum: .
 12.7: Find the following sum: .
 12.8: Find the following sum: .
 12.9: Write the series using sigma notation, then nd its sum:2 + 7 + 12 +...
 12.10: Use mathematical induction to prove that 2 + 4 + 6 + + 2n = n2 + n.
 12.11: Evaluate
 12.12: Find the third term of
 12.13: evaluate the expressions.
 12.14: evaluate the expressions.
 12.15: evaluate the expressions.
 12.16: evaluate the expressions.
 12.17: Expand the expression
 12.18: Use the binomial theorem to expand the binomial
 12.19: Explain why there are always more permutations than combinations.
 12.20: What is the probability of not winning a trifecta (selecting the rs...
 12.21: refer to a roulette wheel with 18 red, 18 black, and 2 green slots....
 12.22: refer to a roulette wheel with 18 red, 18 black, and 2 green slots....
 12.23: refer to a roulette wheel with 18 red, 18 black, and 2 green slots....
 12.24: Marbles. If there are four red marbles,three blue marbles,two green...
 12.25: Cards. What is the probability of drawing an ace or a diamond from ...
 12.26: Human Anatomy. Vasopressin is a relatively simple protein that is f...
 12.27: Find the constant term in the expression
 12.28: Use a graphing calculator to sum a125 n=1C11 4 + 5 6(n  1)D.
 12.1: Simplify
 12.2: The length of a rectangle is 5 less than twice the width and the pe...
 12.3: Solve using the quadratic formula:
 12.4: Solve and express the solution in interval notation: .
 12.5: Write an equation of the line with slope undened and xintercept
 12.6: Find the xintercept and yintercept and slope of the line
 12.7: Using the function evaluate the difference quotient
 12.8: Find the composite function and state the domain for and
 12.9: Find the vertex of the parabola
 12.10: Factor the polynomial as a product of linear factors.
 12.11: Find the vertical and horizontal asymptotes of the function:
 12.12: How much money should be put in a savings account now that earns a ...
 12.13: Evaluate using the changeofbase formula. Round your answer to thr...
 12.14: Solve Round your answer to three decimal places.
 12.15: Solve the system of linear equations. y = 8 5 x + 10 8x  5y = 15
 12.16: Solve the system of linear equations. 2x y z 1 x y 4z 3
 12.17: Maximize the objective function subject to the constraints x + y 5,...
 12.18: Solve the system using GaussJordan elimination.x + 5y  2z = 3 x + ...
 12.19: Given nd C(A B).
 12.20: Calculate the determinant.
 12.21: Find the equation of a parabola with vertex (3, 5) and directrix
 12.22: Graph x2 + y2 6 4.
 12.23: Find the sum of the nite series
 12.24: Classify the sequence as arithmetic, geometric, or neither: 5, 15, ...
 12.25: There are 10 true/false questions on a quiz. Assuming no answers ar...
 12.26: Use a graphing calculator to sum
 12.27: Find the constant term in the expression .
Solutions for Chapter 12: Sequences, Series, and Probability
Full solutions for Algebra and Trigonometry  3rd Edition
ISBN: 9780470648032
Solutions for Chapter 12: Sequences, Series, and Probability
Get Full SolutionsSince 709 problems in chapter 12: Sequences, Series, and Probability have been answered, more than 62581 students have viewed full stepbystep solutions from this chapter. Chapter 12: Sequences, Series, and Probability includes 709 full stepbystep solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9780470648032. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 3.

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).