Discrete Math Quiz
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LOGIC AND PROOF
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A student is proving by mathematical induction that ∑k=1nk3=(n(n+1)2)2\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 for all positive integers nn. In their inductive step, they write: "Assume ∑k=1jk3=(j(j+1)2)2\sum_{k=1}^{j} k^3 = \left(\frac{j(j+1)}{2}\right)^2. Then ∑k=1j+1k3=(j(j+1)2)2+(j+1)3\sum_{k=1}^{j+1} k^3 = \left(\frac{j(j+1)}{2}\right)^2 + (j+1)^3." To complete the proof, they need to show this equals ((j+1)(j+2)2)2\left(\frac{(j+1)(j+2)}{2}\right)^2. Which algebraic identity is most crucial for this verification?
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