Solving Induction Problem: Proving f0f1 + f1f2...f2n-1f2n=f2n2

[bfpri]

So i need to prove f₀f₁ + f₁f₂...f_2n-1f_2n=f_2n² where f_n is the n-th fibonacci number..

Attempting to solve this i let n=k . Then k=1 and f₂²=1 which is 1=1 so that is true.

Then i show if the statement is true for k then it is true for k+1, that is f₀f₁ + f₁f₂+f_2k-1f_2k+f_2kf_2k+1+f_2k+1f_2k+2= f_2k+2² .
I then simplify
f₀f₁ + f₁f₂+f_2k-1f_2k+f_2kf_2k+1+f_2k+1f_2k+2
= f_2k²+ f_2kf_2k+1+f_2k+1f_2k+2.

Now I'm stuck..How do i simplify this to f_2k+2² ?

Thanks

 

[CRGreathouse]

What is f(n) + f(n+1)?

 

[bfpri]

ah, its f(2k+2). Thanks I got it now. Just factor out the f(2k) then factor out the f(2k+2) and you end up with f(2k+2)f(2k+2)

Thanks!