You are going to need to use the defining property of Fibonacci series which is .
Hello I need some help with this problem
For each natural number n, let fn be the nth Fibonacci number. Prove that (f_(n+1))^2 + (f_n)^2 = f_(2n+1).
so far what I have
let n=1 so 1^2 + 1^2 = 2 which is true so the equation holds for n= 1 (base case)
then I pick an arbitrary k where (f_(k+1))^2 + (f_k)^2 = f_(2k+1)
and now I prove for k+1 so I'm trying to show
(f_(k+2))^2 + (f_(k+1)^2 = f_(2k+3)
Ive spent hours trying to manipulate this problem to show this is true but no luck so far so I am hoping someone can guide me through this problem so I know how to do it thanks.
so i substituted and got (f_k + f_(k+1))^2 + (f_(k+1)^2 = f_(2k+3)
what do i do next? do I multiply it out and combine? Substitute in again for some other values?
Note: I have tried many possibilities for this problem, and substituting was one of them, it wasn't as easy as it looks
As often happens with induction, you need to generalize the statement you are proving. Try proving
by induction on m.
Another method is to use matrix multiplication.