- #1
Srumix
- 36
- 0
Homework Statement
Prove that
L1 + 2L2 + 4L3 + 8L4 + ... + 2n-1L1 = 2nFn+1 -1.
Homework Equations
The Fibonacci numbers denoted by Fn is defined as follows:
F1 = 1 , F2 = 1 , F3 = 2 , F4 = 3 , F5 = 5
Fn = Fn-1 + Fn-2
The Lucas numbers are defined as:
Ln = Fn+1 + Fn-1
The Attempt at a Solution
I have attempted to solve this as follows (which I'm sure is wrong):
Assume it's true for n = k and for k + 1:
2kFk+1 -1 + 2k+1Lk+1
=>
2kFk+1 + 2k+1(Fk+2 + Fk)
=>
2k(Fk+1 + 2(Fk+2 + Fk))
But from here, I'm not quite sure how to proceed (if this is in fact, which i doubt, the right way to approach this problem).
Also, note that I'm quite inexperienced when it comes to number theory so this may in fact be a trivially easy problem, but I'm having a hard time to see how to correctly approach these types of problems.
Thanks in advance!