Answer: 4(2 2k - 1) + 3.
- A (2<sup>2k</sup> - 1) + 4
- B 4(2<sup>2k</sup> - 1) + 3
- C 4(2<sup>2k</sup> - 1) - 3
- D 2(2<sup>2k</sup> - 1) + 1
Correct answer: B. 4(2<sup>2k</sup> - 1) + 3
Explanation: 2<sup>2k+2</sup> - 1 = 4 . 2<sup>2k</sup> - 1 = 4(2<sup>2k</sup> - 1) + 3, and since 2<sup>2k</sup> - 1 is divisible by 3 by the inductive hypothesis, the whole expression is divisible by 3.
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.