Question

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Prove that \((1+2+3+\cdots+n)^{2}=1^{3}+2^{3}+3^{3}+\cdots+n^{3}\) for every \(n \in \mathbb{N}\).

Short answer

The base case is proven for n=1. Assuming the statement holds for n=k, it can also be shown to hold for n=k+1 through algebraic manipulation. Thus, by mathematical induction, the formula is true for all natural numbers.

Step-by-step solution

Base Case

Firstly, it must be proven that the statement holds for n=1. In this case, both sides of the equation become 1, so the statement is true for n=1.

Inductive Hypothesis

Assume that the statement is true for a certain k in natural numbers, i.e., \((1+2+3+...+k)^2 = 1^3 + 2^3 + ... + k^3\).

Inductive Step

Now, it must be proven that the statement holds for k+1 if it holds for k. This means proving that \((1+2+3+...+k+(k+1))^2 = 1^3 + 2^3 + ... + k^3 + (k+1)^3\). This can be rewritten as: \((k(k+1)/2 + (k+1))^2 = k(k+1)(2k+1)/6 + (k+1)^3\). Expanding both these expressions, it can be seen that they are indeed equal, thus proving the inductive step.