Mastering the Fibonacci Sequence and the Golden Ratio

The Organic Chemistry Tutor

This detailed mathematics tutorial explores the deep connection between the Fibonacci sequence and the Golden Ratio. It begins by defining the Fibonacci sequence recursively, demonstrating how to generate terms by adding the previous two numbers. The video then guides viewers through an empirical discovery process, calculating the ratios of consecutive terms to show how they converge to the Golden Ratio (approximately 1.618). The video progresses to more advanced algebraic concepts, introducing Binet's Formula for calculating the nth term of the Fibonacci sequence without needing the preceding terms. It also demonstrates how the Fibonacci sequence behaves like a geometric sequence for large values of n. The instructor walks through practical problem-solving examples, such as estimating the 20th term given the 12th term using the Golden Ratio as a multiplier. Finally, the video provides a rigorous mathematical proof deriving the value of the Golden Ratio from the recursive definition of the Fibonacci sequence. By treating the sequence as a geometric progression and solving the resulting quadratic equation (r^2 - r - 1 = 0), the instructor mathematically proves why the Golden Ratio is (1 + ∕5) / 2. This video is an excellent resource for high school algebra, pre-calculus, and calculus classrooms to bridge arithmetic sequences with algebraic proofs.