Let S(n) be a statement parameterized by a positive integer n. A proof by strong induction is used to show that for any n212, S(n) is true. The inductive step shows that for any k215, if S(k-3) is true, then S(k+1) is true.Which fact or set of facts must be proven in the base case of the proof? O (12) O S(15) O S(12), S(13), and S(14) O S(12), S(13), S(14), and S(15)

True or false?

Consider the statement that 3 divides n³ + 2n whenever n is a positive integer. Outline the proof by clicking and dragging to complete each statement. Let P(n) be the proposition that Basis step: P(1) states that Inductive step: Assume that Show that We have completed the basis step and the inductive step. By mathematical induction, we know that vk>0, (P(K)→ P(k+ 1)) is true, that is, vk>0, (3 divides k³ + 2k→ 3 divides (k+ 1)³ + 2(k+ 1)). 3 divided K³ + 2k for an arbitrary integer k> 0. P(n) is true for all integers n ≥ 1. 3 divides 1³ +2 3 divides n³ + 2n. 1, which is true since 1³ +21=3, and 3 divides 3. Reset Click and drag statements to fill in the details of showing that VK(P(K) → P(k+ 1)) is true, thereby completing the induction step. By the inductive hypothesis, 3 divides (k³ + 5k). 3 divides 3(K³ + 1) because it is 3 times an integer. By part (i) of Theorem 1 in Section 4.1, 3 divides the sum (k³ + 2k) + 3(k² + k + 1). This completes the inductive step. (k+ 1)³ + 2(k+ 1) =…

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)

6a. What would the base case be in a proof by induction of this statement? 6b. What would P(k+1) be in a proof by induction of this statement?

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)