Not sure if I'll be able to get a response in time but I'm gonna see what I can get. I have an exam in 3 hours. Even if it's not in time, I'd still like to see how to do this.
Hey I reckon I can prove this, but as a forewarning, I'm definitely no expert in sequences or Fibonacci.
I'll derive everything from F_(k+1) = F_(k) + F_(k-1).
F_(k+1) = F_k + F_(k-1)
(F_(k+1))^2 = (F_k)^2 + 2*F_k*F(k-1) + (F_(k-1))^2 ...Square both sides
(F_(k+1))^2 - (F_k)^2 = 2*F_k*F_(k-1) + (F_(k-1))^2 ...Bring (F_(k))^2 to the left
(F_(k+1))^2 - (F_k)^2 = (F_(k-1))*(2F_k + F_(k-1)) ...Factor out F_(k-2)
(F_(k+1))^2 - (F_k)^2 = (F_(k-1))*(F_k + F_(k+1)) ...Simplify Brackets
After further simplifying brackets:
(F_(k+1))^2 - (F_k)^2 = (F_(k-1))*(F_(k+2))
Ta da!