Prove or disprove: there exists a E N such that for all n E N, an + 1 is prime.

Use induction to prove the following for all natural numbers n: 1 1 1 n (for all n ≥ 1) 3.5 5.7 (2n − 1) (2n + 1) 1.3 + + + + = 2n + 1

Prove that for all integers n, [n/7] = (n – 3)/7 if and only if n mod 7 = 3.

Use the PMI to prove that 3 - 1 is even for all n E N. Proof. Base case: Since 3¹ - 1 = 2, which is even. Thus the statement is true for n = [Select] Inductive step: Assume that there is a natural number n such that 3 - 1 is even. Then 3-1 = [Select] for some integer k. Then 3" [Select] ✓. On both sides, first multiply by 3, then subtract 1, and simplify to get 3+1 -1 = 2( [Select] which is an [Select] integer. Hence, by the PMI, 3 - 1 is even for all n E N. )