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In Relations 1, 2 and 3, we started with a simple term of fibonacci number(s), and repeatedly used the recurrence definition of fibonacci numbers until we observed a pattern. Thus, we could see the particular relation emerging. Relation 4 came up after an attempt to converge the four different terms of Relation 3.
In Relation 5, we first learned the identity in its full. And then for proving it, we used Mathematical Induction. All other relations can also be proven by Mathematical Induction: Relation 1 and 2 by induction over n, Relation 3 by induction over k for any fixed n.
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