Answer: Hence, by the principle of mathematical induction, P(n) is true for all natural numbers n.
- A Hence P(n) is true only for n=1, since that is the only case actually checked
- B Hence, by the principle of mathematical induction, P(n) is true for all natural numbers n
- C Hence P(k) is false, so the inductive step cannot proceed any further from here
- D Hence the proof remains incomplete without further verification of every individual case
Correct answer: B. Hence, by the principle of mathematical induction, P(n) is true for all natural numbers n
Explanation: A correct induction proof concludes by invoking the principle itself: since the base case and inductive step both hold, P(n) is true for all natural numbers n (from the base case onward).
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.