Vijay Maths Tutor (PGCE BSc PhD)

Introduction The Fibonacci sequence is defined using the recurrence formula: f n+2 = f n+1 + f n for n ≧ 1 with f 1 = 1 and f 2 = 1. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... In this post, we explore four elegant results that can be proven using mathematical induction  (including strong induction). 1. Every Fourth Fibonacci Number Is Divisible by 3 Theorem The subsequence f 4n is divisible by 3 for n ≧ 1. Proof (ordinary induction) When

Introduction The sums and have surprisingly elegant closed-form expressions. These identities are extremely useful in A-level/Further Maths trigonometry, complex numbers, and Fourier-type expansions. Below are the two main formulas, followed by two different proofs: a classical trigonometric proof using sum-to-product and telescoping cancellation a short, elegant proof using geometric series and complex numbers. 1. Theorem — Sum of Sines Proof (Trigonometric Method) The resul

The Catenary Curve: Unraveling the Mystery of Hanging Chains