Answer: n = 4.
- A n = 1
- B n = 4
- C n = 0
- D n = 2
Correct answer: B. n = 4
Explanation: The inequality n! > 2<sup>n</sup> is false for n=1,2,3 (check: 1!=1 vs 2, 2!=2 vs 4, 3!=6 vs 8, all fail), but holds from n=4 onward (4!=24 > 16=2<sup>4</sup>). So the base case must be n=4.
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.