Fibonacci Identities

Module to compute (Subscript[F, n]), the nth Fibonacci number:

computeFib[n_]:=Module[{acc0=0,acc1=1,next},
Do[
next=acc0+acc1;  (* compute the next value *)
acc0=acc1;               (* shift down *)
acc1=next,
{k,1,n}
];
acc0         (* return value *)
]

Try it out:

computeFib /@ Range[20]

Note that this version of computeFib also computes (Subscript[F, 0]) correctly:

computeFib[0]

Investigation

Compute (Subscript[F, n]^2)-(Subscript[F, n+1])(Subscript[F, n-1]) for various values of n. What do you observe? What do you wonder?