Question
Using the principle of mathematical induction, show that
Answer :
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We are asked to prove by mathematical induction that [ 1^2 + 3^2 + 5^2 + \dots + (2n-1)^2 = \frac{1}{3} n(4n^2 - 1), \quad \forall n \in \mathbb{N}. ] Step 1: Base Case (n = 1) Left-hand side (LHS): (1^2 = 1) Right-hand side (RHS): (\frac{1}{3} \cdot 1 \cdot (4 \cdot 1^2 - 1) = \frac{1}{3} \cdot (4 - 1) = \frac{3}{3} = 1) LHS = RHS, so the base case holds. Step 2: Induction Hypothesis Assume the formula is true for (n = k),
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