URI CSC 340 F22 HW1 Q8

Base case:
1² = 1(1)(3)/3 ⇔ 1 = 1 ⇔ true

Inductive hypothesis: 1² + 3² + 5² + ... + (2n - 1)² = n(2n - 1)(2n + 1)/3 ⇒ 1² + 3² + 5² + ... + (2n - 1)² + (2(n + 1) - 1)² = (n + 1)(2(n + 1) - 1)(2(n + 1) + 1)/3
⇔ n(2n - 1)(2n + 1)/3 + (2(n + 1) - 1)² = (n + 1)(2(n + 1) - 1)(2(n + 1) + 1)/3
⇔ 4n³ + 12n² + 11n + 3 = 4n³ + 12n² + 11n + 3
⇔ 1 = 1
⇔ true

∴ 1² + 3² + 5² + ... + (2n - 1)² = n(2n - 1)(2n + 1)/3 ∀n ≥ 1