I have been learning about the Fibonacci Numbers and I have been given the task to research on it. I have been assigned to decribe the relationship between the photo (attached below). I know that the relationship is that the "sum of the squares of the first n terms is the nth term multiplied by the (nth+1) term", but I don't think that is worded right? Would this be correct?
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You want to prove that $$ \sum_{k=1}^{n} F_k^2 = F_n F_{n+1}. $$ That is trivial by induction. Equality holds for $n=1$ and: $$ \sum_{k=1}^{n+1} F_k^2 = F_{n+1}^2 +\sum_{k=1}^{n}F_k^2 = F_{n+1}^2 + F_n F_{n+1} = F_{n+1}(F_n+F_{n+1}) = F_{n+1}F_{n+2}.$$
Jack D'Aurizio
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1Yes, but I was wondering how I would explain the relationship (in words) – J. Doe Oct 03 '15 at 10:17