Answer: 7(7 k - 3 k ) + 4(3 k ).
- A 7(7<sup>k</sup> - 3<sup>k</sup>) + 4(3<sup>k</sup>)
- B 7<sup>k</sup> - 3<sup>k</sup> + 4
- C 7 x 7<sup>k</sup> - 3 x 3<sup>k</sup> only, with no simplification
- D 4(7<sup>k</sup> - 3<sup>k</sup>)
Correct answer: A. 7(7<sup>k</sup> - 3<sup>k</sup>) + 4(3<sup>k</sup>)
Explanation: 7<sup>k+1</sup> - 3<sup>k+1</sup> = 7 x 7<sup>k</sup> - 3 x 3<sup>k</sup> = 7(7<sup>k</sup> - 3<sup>k</sup>) + 7(3<sup>k</sup>) - 3(3<sup>k</sup>) = 7(7<sup>k</sup>-3<sup>k</sup>) + 4(3<sup>k</sup>). Since 7<sup>k</sup>-3<sup>k</sup> is divisible by 4 (hypothesis) and 4(3<sup>k</sup>) is clearly divisible by 4, the whole expression is divisible by 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.