Discrete Mathematics With Applications
5th Edition
ISBN: 9781337694193
Author: EPP, Susanna S.
Publisher: Cengage Learning,
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Chapter 5.9, Problem 4ES
To determine
(a)
To find:
The derivations showing that
To determine
(b)
To find:
The derivations showing that
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Chapter 5 Solutions
Discrete Mathematics With Applications
Ch. 5.1 - The notation k=xnnak is read”_________”Ch. 5.1 - The expanded from of k=mnak is _____.Ch. 5.1 - The value of a1+a2+a3x=xn+...+an when n=2 is...Ch. 5.1 - The notation k=mnak is read”______”Ch. 5.1 - If n is a positive integer, then n!=_________Ch. 5.1 - k=nnckck=mnbk=Ch. 5.1 - (k=mnak)(k=mnbk)=Ch. 5.1 - Write the first four terms of the sequences...Ch. 5.1 - Write the first four terms of the sequences...Ch. 5.1 - Write the first four terms of the sequences...
Ch. 5.1 - Write the first four terms of the sequences...Ch. 5.1 - Write the first four terms of the sequences...Ch. 5.1 - Write the first four terms of the sequences...Ch. 5.1 - Let ak=2k+1 and bk=(k1)3+k+2 for every integer k0...Ch. 5.1 - Compute the first fifteen terms of each of the...Ch. 5.1 - Compute the first fifteen terms of each of the...Ch. 5.1 - Find explicit formulas for sequences of the form...Ch. 5.1 - Find explicit formulas for sequences of the from...Ch. 5.1 - Find explicit formulas for sequences of the form...Ch. 5.1 - Find explicit formulas for sequences of the form...Ch. 5.1 - Find explicit formulas for sequences of the form...Ch. 5.1 - Find explicit formulas for sequences of the form...Ch. 5.1 - Find explicit formulas for sequences of the form...Ch. 5.1 - Considser the sequence defined by an=2n+( 1)n14...Ch. 5.1 - Let a0=2,a1=3,a2=2,a3=1,a4=0,a5=1 and a6=2 ....Ch. 5.1 - Compute the summations and products in 19-28....Ch. 5.1 - Compute the summations and products in 19-28....Ch. 5.1 - Compute the summations and products in 19-28....Ch. 5.1 - Prob. 22ESCh. 5.1 - Prob. 23ESCh. 5.1 - Prob. 24ESCh. 5.1 - Compute the summations and products in 19-28....Ch. 5.1 - Compute the summations and products in 19-28....Ch. 5.1 - Compute the summations and products in 19-28....Ch. 5.1 - Compute the summations and products in 19-28....Ch. 5.1 - Prob. 29ESCh. 5.1 - Write the summations in 29-32 in expanded form....Ch. 5.1 - Prob. 31ESCh. 5.1 - Write the summations in 29-32 in expanded form....Ch. 5.1 - Prob. 33ESCh. 5.1 - Evaluate the summations and products in 33-36 for...Ch. 5.1 - Prob. 35ESCh. 5.1 - Prob. 36ESCh. 5.1 - Prob. 37ESCh. 5.1 - Prob. 38ESCh. 5.1 - Prob. 39ESCh. 5.1 - Rewrite 40-42 by separating off the final term....Ch. 5.1 - Rewrite 40-42 by separating off the final term....Ch. 5.1 - Rewrite 40-42 by separating off the final term....Ch. 5.1 - Prob. 43ESCh. 5.1 - Prob. 44ESCh. 5.1 - Prob. 45ESCh. 5.1 - Prob. 46ESCh. 5.1 - Prob. 47ESCh. 5.1 - Prob. 48ESCh. 5.1 - Prob. 49ESCh. 5.1 - Prob. 50ESCh. 5.1 - Prob. 51ESCh. 5.1 - Prob. 52ESCh. 5.1 - Transform each of 53 and 54 by making the change...Ch. 5.1 - Tranfrom each 55-58 by making the change of...Ch. 5.1 - Tranfrom each 55-58 by making the change of...Ch. 5.1 - Transform each of 55-58 by making the change of...Ch. 5.1 - Tranfrom each 55-58 by making the change of...Ch. 5.1 - Tranfrom each 55-58 by making the change of...Ch. 5.1 - Prob. 59ESCh. 5.1 - Write each of 59-61 as a single summation or...Ch. 5.1 - Prob. 61ESCh. 5.1 - Compute each of 62-76. Assume the values of the...Ch. 5.1 - Compute each of 62-76. Assume the values of the...Ch. 5.1 - Compute each of 62-76. Assume the values of the...Ch. 5.1 - Compute each of 62-76 Assume the values of the...Ch. 5.1 - Compute each of 62-76 Assume the values of the...Ch. 5.1 - Compute each of 62-76 Assume the values of the...Ch. 5.1 - Compute each of 62-76. Assume the values of the...Ch. 5.1 - Compute each of 62-76. Assume the values of the...Ch. 5.1 - Compute each of 62-76. Assume the values of the...Ch. 5.1 - Compute each of 62-76. Assume the values of the...Ch. 5.1 - Compute each of 62-76. Assume the valus of the...Ch. 5.1 - Compute each of 62-76. Assume the valus of the...Ch. 5.1 - Compute each of 62-76. Assume the valus of the...Ch. 5.1 - Compute each of 62-76. Assume the valus of the...Ch. 5.1 - Compute each of 62-76. Assume the valus of the...Ch. 5.1 - a. Prove that n!+2 is divisible by 2, for every...Ch. 5.1 - Prove that for all nonnegative integers n and r...Ch. 5.1 - Prove that if p is a prime number and r is an...Ch. 5.1 - Suppose a[1],a[2],a[3],....a[m] is a...Ch. 5.1 - Use repeated division by 2 to convert (by hand)...Ch. 5.1 - Use repeated division by 2 to convert (by hand)...Ch. 5.1 - Prob. 83ESCh. 5.1 - Make a trace table to trace the action of...Ch. 5.1 - Prob. 85ESCh. 5.1 - Prob. 86ESCh. 5.1 - Write an informal description of an algorithm...Ch. 5.1 - Prob. 88ESCh. 5.1 - Prob. 89ESCh. 5.1 - Prob. 90ESCh. 5.1 - Prob. 91ESCh. 5.2 - Mathematical induction is a method for proving...Ch. 5.2 - Prob. 2TYCh. 5.2 - Use the technique illustrated at the beginning of...Ch. 5.2 - For each positive integer n, let P(n) be the...Ch. 5.2 - Fro each positive integer n, let P(n) be the...Ch. 5.2 - For each integer n with n2 , let P(n) be the...Ch. 5.2 - Fill in the missing pieces in the following proof...Ch. 5.2 - Prove each statement in 6-9 using mathematical...Ch. 5.2 - Prove each statement in 6-9 using mathematical...Ch. 5.2 - Prove each statement in 6-9 using mathematical...Ch. 5.2 - Prove each statement in 6-9 using mathematical...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - Prove each of the statements in 10-18 by...Ch. 5.2 - (For students who have Studied calculus) Use...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Use the formula for the sum of the first n...Ch. 5.2 - Prob. 30ESCh. 5.2 - Compute values of the product...Ch. 5.2 - Observe that...Ch. 5.2 - Find a formula in n,a,m, and d for the um...Ch. 5.2 - Find a formaula in a,r,m, and n for the sum...Ch. 5.2 - You have two parents, four grandparents, eight...Ch. 5.2 - Find the mistakes in the proof fragments in 36-38....Ch. 5.2 - Prob. 37ESCh. 5.2 - Theorem: For any interger n1, t=1ni(i!)=(n+1)!1...Ch. 5.2 - Use Theorem 5.2.1 to prove that if m and n are any...Ch. 5.2 - Use Theorem 5.2.1 and the resuly of exercise 10 to...Ch. 5.3 - Mathematical induction differs from the kind of...Ch. 5.3 - Prob. 2TYCh. 5.3 - Use mathematical induction (and the proof of...Ch. 5.3 - Use mathematical induction to show that any...Ch. 5.3 - Prob. 3ESCh. 5.3 - For each positive integer n, let P(n) be the...Ch. 5.3 - For each positive integer n, let P(n) be the...Ch. 5.3 - For each positive integer n, let P(n) be the...Ch. 5.3 - For each positive integer n, let P(n) be the...Ch. 5.3 - Prove each statement in 8—23 by mathematical...Ch. 5.3 - Prove each statement in 8—23 by mathematical...Ch. 5.3 - Prove each statement in 8—23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - Prove each statement in 8-23 by mathematical...Ch. 5.3 - A sequence a1,a2,a3.... is defined by letting a1=3...Ch. 5.3 - A sequence b0,b1,b2... is defined by letting b0=5...Ch. 5.3 - Prob. 26ESCh. 5.3 - A Sequenve d1,d2,d3.... is defined by letting d1=2...Ch. 5.3 - Prove that for every integer n1,...Ch. 5.3 - Exercises 29 and 30 use the definition of string...Ch. 5.3 - Exercises 29 and 30 use the definition of string...Ch. 5.3 - Prob. 31ESCh. 5.3 - Some 55 checkerboards with one square removed can...Ch. 5.3 - Consider a 46 checkerboard. Draw a covering of the...Ch. 5.3 - a. Use mathematical induction to prove that for...Ch. 5.3 - Let m and n be any integers that are greater than...Ch. 5.3 - In a round-robin tournament each team plays every...Ch. 5.3 - On the outside rim of a circular disk the integers...Ch. 5.3 - Suppose that n a’s and nb’s are distributed around...Ch. 5.3 - For a polygon to be convex means that given any...Ch. 5.3 - a. Prove that in an 88 checkerboard with...Ch. 5.3 - Prob. 41ESCh. 5.3 - Prob. 42ESCh. 5.3 - Define a game as follows: You begin with an urn...Ch. 5.3 - Prob. 44ESCh. 5.3 - In order for a proof by mathematical induction to...Ch. 5.3 - In order for a proof by mathematical induction to...Ch. 5.4 - In a proof by strong mathematical induction the...Ch. 5.4 - Prob. 2TYCh. 5.4 - According to the well-ordering principle for the...Ch. 5.4 - Suppose a1,a2,a3,... is a sequence defined as...Ch. 5.4 - Suppose b1,b2,b3,... is a sequence defined as...Ch. 5.4 - Suppose that c0,c1,c2,... is a sequence defined as...Ch. 5.4 - Suppose that d1,d2,d3... is a sequence defined as...Ch. 5.4 - Prob. 5ESCh. 5.4 - Suppose that f0f1,f2... is a sequence defined as...Ch. 5.4 - Suppose that g1,g2,g3,... is a sequence defined as...Ch. 5.4 - Suppose that h0,h1,h2,... is a sequence defined as...Ch. 5.4 - Define a sequence a1,a2,a3,... as follows:...Ch. 5.4 - The introfuctry example solved with ordinary...Ch. 5.4 - You begin solving a jigsaw puzzle by finding two...Ch. 5.4 - The sides of a circular track contain a sequence...Ch. 5.4 - Use strong mathematical induction to prove the...Ch. 5.4 - Any product of two more integers is a result of...Ch. 5.4 - Define the “sum” of one integer to be that...Ch. 5.4 - Use strong mathematical induction to prove that...Ch. 5.4 - Prob. 17ESCh. 5.4 - Compute 9o,91,92,93,94 , and 95 . Make a cojecture...Ch. 5.4 - Suppose that a1,a2,a3,... is a sequence defined as...Ch. 5.4 - Suppose that b1,b2,b3,... is a sequence defined as...Ch. 5.4 - Suppose that c1,c2,c3... is a sequence defined as...Ch. 5.4 - One version of the game NIM starts with two piles...Ch. 5.4 - Define a game G as follows: Begin with a pile of n...Ch. 5.4 - Imagine a situation in which eight people,...Ch. 5.4 - Find the mistake in the following “proof” that...Ch. 5.4 - Use the well-ordering principle for the integers...Ch. 5.4 - Use the well-odering principle fro the integers to...Ch. 5.4 - Prob. 28ESCh. 5.4 - Prob. 29ESCh. 5.4 - Prob. 30ESCh. 5.4 - Prob. 31ESCh. 5.4 - Suppose P(n) is a property such that...Ch. 5.4 - Prove that if a statement can be proved by strong...Ch. 5.4 - It is a fact that every integer n1 can be written...Ch. 5.4 - Prob. 35ESCh. 5.4 - Prove that if a statement can be proved by...Ch. 5.4 - Prob. 37ESCh. 5.5 - A pre-condition for an algorithm is ____ and a...Ch. 5.5 - A loop is defined as correct with respect to its...Ch. 5.5 - Prob. 3TYCh. 5.5 - Prob. 4TYCh. 5.5 - Prob. 1ESCh. 5.5 - Exercises 1-5 contains a while loop and a...Ch. 5.5 - Prob. 3ESCh. 5.5 - Exercise 1-5 conrain a while loop and a predicate....Ch. 5.5 - Exercise 1-5 conrain a while loop and a predicate....Ch. 5.5 - Prob. 6ESCh. 5.5 - Prob. 7ESCh. 5.5 - Exercises 6-9 each contain a while loop annoted...Ch. 5.5 - Prob. 9ESCh. 5.5 - Prob. 10ESCh. 5.5 - Prob. 11ESCh. 5.5 - The following sentence could be added to the loop...Ch. 5.6 - A recursive definition for a sequence consists of...Ch. 5.6 - A recurrence relation is an equation that defines...Ch. 5.6 - Prob. 3TYCh. 5.6 - To solve a problem recurisively means to divede...Ch. 5.6 - Prob. 5TYCh. 5.6 - Find the first four terms every of the recursively...Ch. 5.6 - Find the first four terms of each of the...Ch. 5.6 - Find the first four terms of each of the...Ch. 5.6 - Find the first four terms of each of the...Ch. 5.6 - Find the first four terms of each of the...Ch. 5.6 - Find the first four terms of each of the...Ch. 5.6 - Find the first four terms of each of the...Ch. 5.6 - Find the first four terms of each of the...Ch. 5.6 - Prob. 9ESCh. 5.6 - Let b0,b1,b2... be defined by the formula bn=4n,...Ch. 5.6 - Let c0,c1,c2,... be defined by the formula cn=2n1...Ch. 5.6 - Let S0,S1,S2,... be defined by the formula Sn=(...Ch. 5.6 - Prob. 13ESCh. 5.6 - Let d0,d1,d2,... be defined by the formula dn=3n2n...Ch. 5.6 - For the sequence of Catalan numbers defined in...Ch. 5.6 - Use the recurrence relation and values for the...Ch. 5.6 - Tower of Hanoi with Adjacency Requirement: Suppose...Ch. 5.6 - Prob. 18ESCh. 5.6 - Four-Pole Tower of Hanoi: Suppose that the Tower...Ch. 5.6 - Tower of Hanoi Poles in a Curie: Suppose that...Ch. 5.6 - Double Tower of Hanoi: In this variation of the...Ch. 5.6 - Fibonacci Variation: A single pair of rabbits...Ch. 5.6 - Fibonacci Variation: A single pair of rabbits...Ch. 5.6 - In 24-34, Fa,F1,F2,...is the Fibonacci sequence....Ch. 5.6 - In 24-34, Fa,F1,F2,...is the Fibonacci sequence....Ch. 5.6 - In 24—34, F0,F1,F2,.... is the Fibonacci sequence....Ch. 5.6 - Prob. 27ESCh. 5.6 - Prob. 28ESCh. 5.6 - Prob. 29ESCh. 5.6 - Prob. 30ESCh. 5.6 - In 24-34, Fa,F1,F2,...is the Fibonacci sequence....Ch. 5.6 - In 24-34, Fa,F1,F2,...is the Fibonacci sequence....Ch. 5.6 - Prob. 33ESCh. 5.6 - Prob. 34ESCh. 5.6 - Prob. 35ESCh. 5.6 - Prob. 36ESCh. 5.6 - Prob. 37ESCh. 5.6 - Compound Interest: Suppose a certain amount of...Ch. 5.6 - With each step you take when climbing a staircase,...Ch. 5.6 - A set of blocks contains blocks of heights 1, 2,...Ch. 5.6 - Prob. 41ESCh. 5.6 - Prob. 42ESCh. 5.6 - Prob. 43ESCh. 5.6 - Prob. 44ESCh. 5.6 - Prob. 45ESCh. 5.6 - Prob. 46ESCh. 5.6 - Prob. 47ESCh. 5.7 - To use iteration to find an explicit formula for a...Ch. 5.7 - At every step of the iteration process, it is...Ch. 5.7 - If a single number, say a, is added to itself k...Ch. 5.7 - If a single number, say a, is multiplied by itself...Ch. 5.7 - A general arithmetic sequence a0,a1,a2,... with...Ch. 5.7 - Prob. 6TYCh. 5.7 - Prob. 7TYCh. 5.7 - The formula 1+2+3++n=n(n+1)2 is true for every...Ch. 5.7 - The formula 1+r+r2++rn=rn+11r1 is true for every...Ch. 5.7 - In each of 3—15 a sequence is defined recursively....Ch. 5.7 - In each of 3—15 a sequence is defined recursively....Ch. 5.7 - In each of 3-15 a sequence is defined recursively....Ch. 5.7 - In each of 3-15 a sequence is defined recursively....Ch. 5.7 - Prob. 7ESCh. 5.7 - In each of 3-15 a sequence is defined recursively....Ch. 5.7 - In each of 3-15 a sequence is defined recursively....Ch. 5.7 - Prob. 10ESCh. 5.7 - In each of 3-15 a sequence is defined recursively....Ch. 5.7 - In each of 3-15 a sequence is defined recursively....Ch. 5.7 - Prob. 13ESCh. 5.7 - In each of 3-15 a sequence is defined recursively....Ch. 5.7 - In each of 3-15 a sequence is defined recursively....Ch. 5.7 - Solve the recurrence relation obtained as the...Ch. 5.7 - Solve the recurrence relation obtained as the...Ch. 5.7 - Prob. 18ESCh. 5.7 - A worker is promised a bonus if he can increase...Ch. 5.7 - Prob. 20ESCh. 5.7 - Prob. 21ESCh. 5.7 - As shown in Example 5.6.8, if a bank pays interest...Ch. 5.7 - Prob. 23ESCh. 5.7 - A chain letter works as follows: One person sends...Ch. 5.7 - A certain computer algorithm executes twice as...Ch. 5.7 - A person saving for retirement makes an initial...Ch. 5.7 - A person borrows $3,000on a bank credit card at a...Ch. 5.7 - Prob. 28ESCh. 5.7 - In 28-42 use mathematical induction to verify the...Ch. 5.7 - In 28-42 use mathematical induction to verify the...Ch. 5.7 - Prob. 31ESCh. 5.7 - In 28-42 use mathematical induction to verify the...Ch. 5.7 - Prob. 33ESCh. 5.7 - Prob. 34ESCh. 5.7 - In 28-42 use mathematical induction to verify the...Ch. 5.7 - Prob. 36ESCh. 5.7 - In 28-42 use mathematical induction to verify the...Ch. 5.7 - In 28-42 use mathematical induction to verify the...Ch. 5.7 - Prob. 39ESCh. 5.7 - Prob. 40ESCh. 5.7 - In 28-42 use mathematical induction to verify the...Ch. 5.7 - Prob. 42ESCh. 5.7 - Prob. 43ESCh. 5.7 - In each of 43-49 a sequence is defined...Ch. 5.7 - In each of 43-49 a sequence is defined...Ch. 5.7 - Prob. 46ESCh. 5.7 - Prob. 47ESCh. 5.7 - In each of 43—49 a sequence is defined...Ch. 5.7 - Prob. 49ESCh. 5.7 - Prob. 50ESCh. 5.7 - In 50 and 51 determine whether the given...Ch. 5.7 - A single line divides a plane into two regions....Ch. 5.7 - Compute [ 1 101]n for small values of n(up to...Ch. 5.7 - Prob. 54ESCh. 5.8 - A second-order linear homogeneous recurrence...Ch. 5.8 - Prob. 2TYCh. 5.8 - Prob. 3TYCh. 5.8 - If a sequence a1,a2,a3,... is defined by a...Ch. 5.8 - Which of the following are second-order linear...Ch. 5.8 - Which of the following are second-order linear...Ch. 5.8 - Let a0,a1,a2,.... be the sequence defined by the...Ch. 5.8 - Let b0,b1,b2,... be the sequence defined by the...Ch. 5.8 - Let a0,a1,a2,... be the sequence defined by the...Ch. 5.8 - Let b0,b1,b2... be the sequence defined by the...Ch. 5.8 - Solve the system of equations in Example 5.8.4 to...Ch. 5.8 - In each of 8—10: (a) suppose a sequence of the...Ch. 5.8 - In each of 8—10: (a) suppose a sequence of the...Ch. 5.8 - In each of 8-10: (a) suppose a sequence of the...Ch. 5.8 - In each of 11-16 suppose a sequence satisfies the...Ch. 5.8 - In each of 11-16 suppose a sequence satisfies the...Ch. 5.8 - Prob. 13ESCh. 5.8 - Prob. 14ESCh. 5.8 - Prob. 15ESCh. 5.8 - In each of 11-16 suppose a sequence satisfies the...Ch. 5.8 - Prob. 17ESCh. 5.8 - Prob. 18ESCh. 5.8 - Prob. 19ESCh. 5.8 - Prob. 20ESCh. 5.8 - Prove Theorem 5.8.5 for the case where the values...Ch. 5.8 - Prob. 22ESCh. 5.8 - Prob. 23ESCh. 5.8 - Prob. 24ESCh. 5.9 - The base for a recursive definition of a set is...Ch. 5.9 - Prob. 2TYCh. 5.9 - Prob. 3TYCh. 5.9 - One way to show that a given element is in a...Ch. 5.9 - Prob. 5TYCh. 5.9 - Prob. 6TYCh. 5.9 - Prob. 1ESCh. 5.9 - Prob. 2ESCh. 5.9 - Prob. 3ESCh. 5.9 - Prob. 4ESCh. 5.9 - Prob. 5ESCh. 5.9 - Prob. 6ESCh. 5.9 - Prob. 7ESCh. 5.9 - Prob. 8ESCh. 5.9 - Define a set S of strings over the set {a, b}...Ch. 5.9 - Prob. 10ESCh. 5.9 - Prob. 11ESCh. 5.9 - Prob. 12ESCh. 5.9 - Define a set S of integers recursively as follows:...Ch. 5.9 - Prob. 14ESCh. 5.9 - Determine wheteher either of the following...Ch. 5.9 - Prob. 16ESCh. 5.9 - Give a recursive definition for the set of all...Ch. 5.9 - Prob. 18ESCh. 5.9 - Give a recursive definition for the set all...Ch. 5.9 - a. Let A be any finite set let L be the length...Ch. 5.9 - Prob. 21ESCh. 5.9 - Prob. 22ESCh. 5.9 - Use the definition of McCarthy’s 91 function in...Ch. 5.9 - Prove that McCarthy’s 91 function equals 91 for...Ch. 5.9 - Use the definition of the Ackermann function in...Ch. 5.9 - Prob. 26ESCh. 5.9 - Prob. 27ESCh. 5.9 - Prob. 28ESCh. 5.9 - Prob. 29ES
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- use L'Hopital Rule to evaluate the following. a) 4x3 +10x2 23009׳-9 943-9 b) hm 3-84 хто бу+2 < xan x-30650)arrow_forwardConstruct a know-show table for each statement below that appears to be true.arrow_forwardProblem 3. Pricing a multi-stock option the Margrabe formula The purpose of this problem is to price a swap option in a 2-stock model, similarly as what we did in the example in the lectures. We consider a two-dimensional Brownian motion given by W₁ = (W(¹), W(2)) on a probability space (Q, F,P). Two stock prices are modeled by the following equations: dX = dY₁ = X₁ (rdt+ rdt+0₁dW!) (²)), Y₁ (rdt+dW+0zdW!"), with Xo xo and Yo =yo. This corresponds to the multi-stock model studied in class, but with notation (X+, Y₁) instead of (S(1), S(2)). Given the model above, the measure P is already the risk-neutral measure (Both stocks have rate of return r). We write σ = 0₁+0%. We consider a swap option, which gives you the right, at time T, to exchange one share of X for one share of Y. That is, the option has payoff F=(Yr-XT). (a) We first assume that r = 0 (for questions (a)-(f)). Write an explicit expression for the process Xt. Reminder before proceeding to question (b): Girsanov's theorem…arrow_forward
- Problem 1. Multi-stock model We consider a 2-stock model similar to the one studied in class. Namely, we consider = S(1) S(2) = S(¹) exp (σ1B(1) + (M1 - 0/1 ) S(²) exp (02B(2) + (H₂- M2 where (B(¹) ) +20 and (B(2) ) +≥o are two Brownian motions, with t≥0 Cov (B(¹), B(2)) = p min{t, s}. " The purpose of this problem is to prove that there indeed exists a 2-dimensional Brownian motion (W+)+20 (W(1), W(2))+20 such that = S(1) S(2) = = S(¹) exp (011W(¹) + (μ₁ - 01/1) t) 롱) S(²) exp (021W (1) + 022W(2) + (112 - 03/01/12) t). where σ11, 21, 22 are constants to be determined (as functions of σ1, σ2, p). Hint: The constants will follow the formulas developed in the lectures. (a) To show existence of (Ŵ+), first write the expression for both W. (¹) and W (2) functions of (B(1), B(²)). as (b) Using the formulas obtained in (a), show that the process (WA) is actually a 2- dimensional standard Brownian motion (i.e. show that each component is normal, with mean 0, variance t, and that their…arrow_forwardRoedel Electronics produces tablet computer accessories, including integrated keyboard tablet stands that connect a keyboard to a tablet device and holds the device at a preferred angle for easy viewing and typing. Roedel produces two sizes of integrated keyboard tablet stands, small and large. Each size uses the same keyboard attachment, but the stand consists of two different pieces, a top flap and a vertical stand that differ by size. Thus, a completed integrated keyboard tablet stand consists of three subassemblies that are manufactured by Roedel: a keyboard, a top flap, and a vertical stand. Roedel's sales forecast indicates that 7,000 small integrated keyboard tablet stands and 5,000 large integrated keyboard tablet stands will be needed to satisfy demand during the upcoming Christmas season. Because only 500 hours of in-house manufacturing time are available, Roedel is considering purchasing some, or all, of the subassemblies from outside suppliers. If Roedel manufactures a…arrow_forwardShow three different pairs of integers, a and b, where at least one example includes a negative integer. For each of your examples, determine if each of the following statements are true or falsearrow_forward
- The scores of 8 students on the midterm exam and final exam were as follows. Student Midterm Final Anderson 98 89 Bailey 88 74 Cruz 87 97 DeSana 85 79 Erickson 85 94 Francis 83 71 Gray 74 98 Harris 70 91 Find the value of the (Spearman's) rank correlation coefficient test statistic that would be used to test the claim of no correlation between midterm score and final exam score. Round your answer to 3 places after the decimal point, if necessary. Test statistic: rs =arrow_forward(a) Develop a model that minimizes semivariance for the Hauck Financial data given in the file HauckData with a required return of 10%. Assume that the five planning scenarios in the Hauck Financial rvices model are equally likely to occur. Hint: Modify model (8.10)-(8.19). Define a variable d, for each scenario and let d₂ > R - R¸ with d ≥ 0. Then make the objective function: Min Let FS = proportion of portfolio invested in the foreign stock mutual fund IB = proportion of portfolio invested in the intermediate-term bond fund LG = proportion of portfolio invested in the large-cap growth fund LV = proportion of portfolio invested in the large-cap value fund SG = proportion of portfolio invested in the small-cap growth fund SV = proportion of portfolio invested in the small-cap value fund R = the expected return of the portfolio R = the return of the portfolio in years. Min s.t. R₁ R₂ = R₁ R R5 = FS + IB + LG + LV + SG + SV = R₂ R d₁ =R- d₂z R- d₂ ZR- d₁R- d≥R- R = FS, IB, LG, LV, SG, SV…arrow_forwardThe Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas. The following is a linear program used to determine which cities Martin-Beck should construct a plant in. Let y₁ = 1 if a plant is constructed in Detroit; 0 if not y₂ = 1 if a plant is constructed in Toledo; 0 if not y₂ = 1 if a plant is constructed in Denver; 0 if not y = 1 if a plant is constructed in Kansas City; 0 if not. The variables representing the amount shipped from each plant site to each distribution center are defined just as for a transportation problem. *,, = the units shipped in thousands from plant i to distribution center j i = 1 (Detroit), 2 (Toledo), 3 (Denver), 4 (Kansas City), 5 (St.Louis) and…arrow_forward
- Consider the following mixed-integer linear program. Max 3x1 + 4x2 s.t. 4x1 + 7x2 ≤ 28 8x1 + 5x2 ≤ 40 x1, x2 ≥ and x1 integer (a) Graph the constraints for this problem. Indicate on your graph all feasible mixed-integer solutions. On the coordinate plane the horizontal axis is labeled x1 and the vertical axis is labeled x2. A region bounded by a series of connected line segments, and several horizontal lines are on the graph. The series of line segments connect the approximate points (0, 4), (3.889, 1.778), and (5, 0). The region is above the horizontal axis, to the right of the vertical axis, and below the line segments. At each integer value between 0 and 4 on the vertical axis, a horizontal line extends out from the vertical axis to the series of connect line segments. On the coordinate plane the horizontal axis is labeled x1 and the vertical axis is labeled x2. A region bounded by a series of connected line segments, and several…arrow_forwardConsider the nonlinear optimization model stated below. Min s.t. 2x²-18x + 2XY + y² - 14Y + 53 x + 4Y ≤ 8 (a) Find the minimum solution to this problem. |at (X, Y) = (b) If the right-hand side of the constraint is increased from 8 to 9, how much do you expect the objective function to change? Based on the dual value on the constraint X + 4Y ≤ 8, we expect the optimal objective function value to decrease by (c) Resolve the problem with a new right-hand side of the constraint of 9. How does the actual change compare with your estimate? If we resolve the problem with a new right-hand-side of 9 the new optimal objective function value is| , so the actual change is a decrease of rather than what we expected in part (b).arrow_forwardStatement:If 2 | a and 3| a, then 6 a. So find three integers, and at least one integer should be negative. For each of your examples, determine if the statement is true or false.arrow_forward
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