What is the consequence of not proving a base case in mathematical induction? For example:
prove that 2 + 4 + · · · + 2n = (n + 2)(n − 1) for any integer n ≥ 2
if a student jumped right into the inductive step without proving the base case We assume that P(n) : 2 + 4 + · · · + 2n = (n + 2)(n − 1), and we want to prove P(n + 1) : 2 + 4 + · · · + 2n + 2(n + 1) = (n + 3)n. 2 + 4 + · · · + 2n + 2(n + 1) = 2 + 4 + · · · + 2n + 2n + 2 = (n + 2)(n − 1) + 2n + 2 hypothesis = n(n + 2) − (n + 2) + 2n + 2 = n(n + 2) + n = n(n + 3)
What does this say?