Problem 3: For the following problem state the assumption (what is given and what is it that you aregoing to prove)ti start the proof with if you select each of the a,b,c proving techniques:Problem: If x>5 and y is even then x*y +3 will not be divisible by 2.a) Direct Proof:b) Contrapositive Proof:c) Contradiction Proof:

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Answered: Problem 3: For the following problem…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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for question 2 A,B,C,D please explain
6. In this problem, you may use the fact (which we will prove in Chapter 6) that an integer n is not divisible by 3 if and only if there exists an integer k such that n = 3k +1 or n = 3k + 2. (a) Prove that for all integers n, if 3 | n2, then 3 | n. (b) Prove that for all integers i and j, if 3 | (i2 + j?), then 3 | i and 3 | j.
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Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) 2.) (a) In how many ways can a hand of 6 hearts be chosen from an ordinary deck? (b) In how many ways can a hand of 6 black cards be chosen from an ordinary deck?
24. Binomial Multiplication Patterns in Pascal's Triangle a.) What is (x+1)°? (Hint: anything to the 0 power is b.) What is (x+1)'? c.) What is (x+1)²? Caution: it is not x2+12, but is gotten by multiplying (x + 1)(x + 1). d.) Where can the coefficients and constants of each answer be found in Pascal's triangle? The coefficient is the number in front of x, and the constant is the added on number. For x + 1, the coefficient in front of the x is a 1, so we have 1x + 1. Where do you see 1 1 in the triangle? Where do you see the coefficients and constants for your answer to part c? 4 1 10 10 1 6 I 15 I 20 I 15 7 I 21 I 35 56 I 70 I 56 I 28 36 I 84 1126 | 126 I 84 I 36 1 6 1 1 35 I 21 I 7 1 8 28 8 1 9 e.) What is (x+1)³? Hint: multiply the result of part c by (x+1), then combine like terms. Explain how you know you are correct using Pascal's triangle. f.) Make a conjecture as to what (x+1)4 will equal, using Pascal's triangle. You do not have to multiply it all out to check, but instead,…
Problem 5.4 (Grade a "Proof"). Study the following claim as well as the "proof": Claim. For all n e N, 3| (n³ +44n). “Proof". When n = 1, n³ +44n = 45 is divisible by 3. When n = 2, n³ +44n = 96 is divisible by 3. When n = 3, n³ + 44n = 159 is divisible by 3. When n = 100, n3 + 44n = 1004400 is divisible by 3. Thus 3 | (n³ +44n) for all n e N. %3D Complete the following questions concerning the above claim and "proof": (1) Determine whether the "proof" is rigorous. Identify the issues in the "proof", if any. (2) Determine whether the claim is true or false. Justify the answer in part (3). (3) If the the claim is true and the “proof" is not rigorous, then provide a correct and rigorous proof. If the claim is false, give a concrete counterexample. Hint. Complete the parts as instructed. If the claim is false, then (obviously) there is no way the "proof" could be correct.
Put the steps of a proof for the following claim in the proper order: 4 + 8 + 12 + + 4n = 2n(n + 1) + 4k + 4(k + 1) = 2k (k + 1) +4(k + 1) 2 (k + 1) (k +2) • 4+8+ 12 + .. + 4k = 2k (k + 1). Assume 4 + 8 + 12 + v Let n = 1. Clearly, 4 = 2(1)(1+ 1). v WTS: 4 +8+ 12 + + 4k + 4(k + 1) = 2 (k + 1) ((k + 1) + 1)

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