URI CSC 340 F22 HW1 Q7

Base case:
1/(1 * 2) = 1/(1 + 1) ⇔ 1 = 1 ⇔ true

Inductive hypothesis: 1/(1 * 2) + 1/(2 * 3) + ... + 1/(n * (n + 1)) = n/(n + 1) ⇒ 1/(1 * 2) + 1/(2 * 3) + ... + 1/(n * (n + 1)) + 1/((n + 1) * (n + 2)) = (n + 1)/(n + 2)
⇔ n/(n + 1) + 1/((n + 1) * (n + 2)) = (n + 1)/(n + 2)
(n = -2 is outside the domain of our problem)
⇔ (n² + 2n)/(n + 1) + 1/(n + 1) = (n + 1)
⇔ (n² + 2n) + 1 = (n + 1)²
⇔ n² + 2n + 1 = n² + 2n + 1
⇔ 1 = 1
⇔ true

∴ 1/(1 * 2) + 1/(2 * 3) + ... + 1/(n * (n + 1)) = n/(n + 1) ∀n ≥ 1