Im having some trouble figuring this out.. could someone help me out a bit?
Define the sequence of F as follows
F1=1, F2=1, Fn=Fn-1+Fn-2 n>=3
prove F1^2+F2^2+...Fn^2=FnFn+1 while n is a positive integer
That is, of course, the Fibonacci series. The problem you have is that you are trying to prove something that isn't true! Have you tried proving this by induction? When n= 1, this says F1^2+ F2^2= 1+ 1= 2= F2F3= 1*2= 2.
If it is true that $\displaystyle F_1^2+ ...+ F_k^2= F_kF_{k+1}$, for some k, then $\displaystyle F_1^2+ ...+ F_k^2+ F_{k+1}^2= F_kF_{k+1}+ F_{k+1}^2$.