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

[ghostskwid]

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.

 

[mfb]

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.