Let's look at φ^1, now! Anything to the first power is just that number, so φ^1=φ! Think of that as φ^1 = φ + 0
(because adding 0 does nothing)
Remember our Fibonacci numbers, now. Starts out with 0, 1, 1, 2, etc. Remember that F_0=0 and F_1 = 1?
φ^1 = φ + 0
can be rewritten as
φ^1 = F_1 * φ + F_0
Which means that φ^1 follows that formatting that we used earlier! Thus, every φ^n follows the format
φ^n = F_n * φ^n + F_(n-1)
Anyways I'll stop being a Big Nerd for like 2 minutes
If φ^n = F_n * φ + F_(n-1), then
φ^(n+1) = φ * φ^n
φ^(n+1) =φ * (F_n * φ + F_(n-1))
φ^(n+1) = F_n*φ^2+F_(n-1) * φ
φ^(n+1) = F_n*(φ+1)+F_(n-1) * φ
φ^(n+1) = F_n * φ + F_n + F_(n-1) * φ
φ^(n+1) = (F_n + F_(n-1)) φ + F_n
φ^(n+1) = F_(n+1) φ + F_n
*wipes sweat from brow*
alright, that part's out of the way. the big takeaway there is that if that original property holds for a single φ^n, then it holds for every φ^n after it.
Queer sheep boy. Leftist as hell. I like video games & math. ask for my telegram yo
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