Golden Threeway - Fibonacci, Lucas and the Golden Ratio

The Golden Ratio is closely tied to the sequence of Fibonacci Numbers - but this relationship isn't completely described without involving Lucas Numbers. In this video, we explore the many ways they define and connect with each other. 00:00 Intro 00:45 Recurrence Comparison 02:40 Convergence to Φ 05:20 Finding Fibonacci in Φ 07:21 Φ, ψ and the n-th Fibonacci 10:27 Lucas Numbers 13:08 Fibonacci and Lucas off by a Factor 14:43 n+1, n-1 16:27 Linear Combination Using Lucas Numbers 17:35 All Together Now 18:17 Bonus Equality 20:05 Bonus Graph