I have wrestled with the following two problems for a couple of hours each and have been successful. Now I am interested in how a more experienced mathematician would go about solving these. I encourage you to look past my lack of latex skills (lol), and do your best with the attachment I poorly put together. A note about the second problem, If anyone is unaware F denotes the Fibonacci numbers and L denotes the Lucas numbers. Second, the term on the left side is a little blurry but it says the 2n Fibonacci number. Also I challenge you to solve this second problem without strong induction. Which I semi-managed to do. I proved a secondary result with strong induction then used that result to use regular induction on the main problem.