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