While proving the sum of cubes formula [n(n+1)/2] 2 by induction, the inductive step must show that [k(k+1)/2] 2 + (k+1) 3 equals:

While proving the sum of cubes formula…

Answer: [(k+1)(k+2)/2] 2.

A [(k+1)(k+2)/2]2
B [k(k+2)/2]2
C (k+1)2 (k+2)2
D k2(k+1)2/4 + (k+1)

Correct answer: A. [(k+1)(k+2)/2]2

Explanation: Factor out (k+1)2: [k(k+1)/2]2 + (k+1)3 = (k+1)2 [k2/4 + (k+1)] = (k+1)2 [(k2+4k+4)/4] = (k+1)2 (k+2)2/4 = [(k+1)(k+2)/2]2, matching the formula for n=k+1.

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.

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While proving the sum of cubes formula [n(n+1)/2]<sup>2</sup> by induction, the inductive step must show that [k(k+1)/2]<sup>2</sup> + (k+1)<sup>3</sup> equals:

Concept context

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