Exercise 5.52 asked for a proof of the following statement: For each integer n > 12, there exist nonnegativeintegers a and b such that n = 3a + 7b. Use the Strong Principle of Mathematical Induction to give analternative proof of this statement.

The following two steps are used to prove a statement P(n) by the strong principle of mathematical…

Answered: Exercise 5.52 asked for a proof of the…

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 parts a and d
Discrete math.
Use induction to prove that for every integer n ≥ 0, 5n+2+ 62n+1 is divisible by 31.
I would like to have a clear explanation of the following math induction proof! Thanks! (a) Show that 1+ 2 21+, ... 3 2n-1 27 whenever n is a nonnegative integer.
Need to be done part c and d only
Use the principle of mathematical induction to show that the statement is true for all natural numbers. 2² +4² +6² + ... + (2n)² = 2n(n + 1)(2n + 1) 3 Let Pn denote the statement: 22 +4² +6² + ... + (2n)² + Check that P₁ is true: 2² = 4 and 2(C Assume Pk is true: 22 +42 +62 + + (² 2 (2n)² = 2n(n + 1)(2n + 1) 3 2² +4² +6² + ... + (2(k+1))² ))( + ¹)(² 1) (2 1 3 To show that Pk+1 is true, add (2(k + 1))² to both sides of Pk. 2² + 4² + 6² + ... + (2k)² + (2( ))²³ = 2 2 |)²-² = +1 1) 2k(k + 1)(2k + 1) 12( + 3 = Rewrite the right-hand side as a single fraction, and then factor the numerator completely. 3 3 2k(k + 1)(2k + 1) +(2( 3 (k + 1)(2k + 1) Thus P₁ is true. 2 2 :))²
Consider the statement: For any positive integer nn, n3+2nn3+2n is a multiple of 33. Set up the framework for a proof by mathematical induction by doing the following steps: State the base case, and prove that the base case is true. Clearly state the inductive hypothesis. Clearly state the inductive step (that is, clearly state what needs to be proven following the inductive hypothesis). Give at least one suggestion for how a full proof of the inductive step might go, that is reasonable and likely to be useful.
Write a proof of the following statement using PMI (Principle of Mathematical Induction). Prove all that for all natural numbers n: (see attached screenshot for the problem)

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Exercise 5.52 asked for a proof of the following statement: For each integer n > 12, there exist nonnegative integers a and b such that n = 3a+7b. Use the Strong Principle of Mathematical Induction to give an alternative proof of this statement.