Prove each statement in 6-9 using mathematical induction. Do not derive them from Theorem 5.2.1 or Theorem 5.2.2 For every integer n >=3, 43 + 44 + 45 + … + 4n = 4(4n-16)/3  Show that P(3) is true: 43 = 64 = 4(4n-16)/3 = 192/3 = 64 = 43 If P(k) is true then P(k + 1) is also true: 43 + 44 + 45 + … + 4k = 4(4k-16)/3  43 + 44 + 45 + … + 4k+1= 4(4k+1-16)/3

Use mathematical induction to prove

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) =…

Use induction to prove the following proposition: 1² +2²+3² +4² + . +n² n(n + 1) (2n + 1) 6 VnE I+