QUESTION 10a. Provide two (2) situations where induction method is not applicable?b.By using mathematical induction method, prove that the following statement is true forneZ+.(1)(4) + (2) (5) + (3)(6) + ... + (n)(n+ 3) = n(n+1)(n+ 5).

We first provide two situations where induction method is not applicable.

Answered: a. Provide two (2) situations where…

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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Prove by induction: 1) 2 +4 + 6+ 2) 1 +4 +7+ + 2n = n (n + 1) + (3n – 2) = 1/2 n ( 3n - 1) ....
Given that P(n) is the equation 1·(1!)+2· (2!) + · an integer such that n > 1, we will prove that P(n) is true for all n > 1 by induction. 6. +n· (n!) = (n + 1)! – 1, where n is (a) Base case: i. Write P(1). ii. Show that P(1) is true. In this case, this requires showing that a left-hand side is equal to a right-hand side. (b) Inductive hypothesis: Let k > 1 be a natural number. Assume P(k) is true. Write P(k). (c) Inductive step: i. Write P(k + 1). ii. Use the assumption that P(k) is true to prove that P(k+1) is true. (d) Explain why this shows that P(n) is true for all n > 1.
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Given that P(n) is the equation 1+3+5+7+...+(2n − 1) = n², where n is an integer such that n ≥ 1, we will prove that P(n) is true for all n ≥ 1 by induction. Base case: i. Write P(1). ii. Show that P(1) is true. In this case, this requires showing that a left-hand side is equal to a right-hand side. (b) Inductive hypothesis: Let k ≥ 1 be a natural number. Assume that P(k) is true. Write P(k). (c) Inductive step: i. Write P(k+1). ii. Use the assumption that P(k) is true to prove that P(k+1) is true. Justify all of your steps.
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Which of the following is the denial of (3!x)P(x) OA. ((VX)[P(x)=(3y)(P(y) ^ x =y)}p. B. ((Vx)(P(x) v (X)(¬P(x))). C. X)(¬P(x) v (3y)(32)(y=ZA P(y) A P(Z)))- D.((3x)(P(x)) ^ (y)[¬P(y)=xy)- E. ((3x)(P(x)) ^ (vy)[P(y)=x=y).
I need help in solving the second part of the question specifically using Induction given that the P(n) is (n(n+1)(7+2n))/6
2. Use the binomial theorem to show that for all integers n 2 1, 3" = (6) + 2 (4) + 2° (") ++ 2* (C) +.. · + 2"

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a. Provide two (2) situations where induction method is not applicable? b. By using mathematical induction method, prove that the following statement is true for ne Zt. (1)(4) + (2) (5) + (3) (6) + ... + (n)(n+3) = n(n+1)(n+ 5).