Can You Find a Counterexample to the Recursive Lucas and Fibonacci Relationship?

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The discussion revolves around a relationship involving the sums of the Lucas sequence, expressed as L0 + L1 + L2 + ... + Ln = Ln+2 - 1. The poster seeks a counterexample to this formula using an inductive approach, acknowledging difficulty in proving their findings definitively. They mention that discovering a counterexample would invalidate the formula, regardless of how it was derived. Another participant suggests that there may be a similar relationship that is independent of the starting values, requiring only a modification of the constant in the formula. The conversation emphasizes the importance of proving or disproving recursive relationships in mathematical sequences.
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Hi I am playing around with recursive definitions of Lucas and Fibonacci sequences:

I came across a relationship

L0 + L1 + L2 + L3 ... Ln = sum(i = 0, n) Li = Ln+2 -1;

Sorry for the horrible notation, but could anyone provide a counter example using an inductive approach? I get the counter example through guessing, but am having a hard time proving it definitively.
 
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Any counterexample would show that your formula is wrong, it does not matter how you got that counterexample.

You can show this formula via induction, this is an easy example of induction.
Actually, there should be a similar relation independent of the starting values, where just the constant in the formula has to be changed.
 

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