(a) Use induction to show that n 2 − n is always even. (b) Give a direct proof in one or two sentences that n 2 − n is always even. (c) Show that n 3 − n is always divisible by three. (d) Is n 5 − n...

(a) Use induction to show that n²
− n is always even.

(b) Give a direct proof in one or two sentences that n²
− n is always even.

(c) Show that n³
− n is always divisible by three.

(d) Is n⁵
− n aways divisible by 5? Explain your answer.

Prove that the sum of the first n even numbers is n
²
+ n

(a) indirectly by assuming that the sum of the first n odd numbers is n
²
.

(b) directly by mathematical induction.