Answer: Without a verified base case, the chain of implications has no starting point to begin from.
- A Without a verified base case, the chain of implications has no starting point to begin from
- B The inductive hypothesis must always be assumed false, not true, for the argument to work
- C The statement P(n) must instead be proven separately for every even and odd value of n
- D Strong induction is required instead, since ordinary induction cannot use implications
Correct answer: A. Without a verified base case, the chain of implications has no starting point to begin from
Explanation: Mathematical induction requires both a true base case and a valid inductive step; without an established starting point P(1), the chain of implications P(k) => P(k+1) never gets triggered for any n.
Mathematical induction works like a row of dominoes: proving the base case P(1) tips the first domino, and proving the inductive step (P(k) ⟹ P(k+1)) guarantees each domino knocks over the next - together these two facts guarantee ALL dominoes fall, without checking each one individually.
Concept context
A proof technique used to establish that a statement is true for every natural number, using a base case and an inductive step.