Hello, smoothi963!
We all assume that these are Fibonacci Numbers.
Use induction to prove:
. . (F1)² + (F2)² + (F3)² + … + (Fn)² .= .(Fn)·(Fn+1)
I assume you've verified statement S(1) already.
Assume statement S(k):
. . (F1)² + (F2)² + (F3)² + ... + (Fk)² .= .(Fk)·(Fk+1)
Add (Fk+1)² to both sides:
. . (F1)² + (F2)² + (F3)² + ... + (Fk+1)² .= .(Fk)·(Fk+1) + (Fk+1)²
The LHS is the left side of S(k+1).
Factor the right side: .Fk+1·(Fk + Fk+1) . = .(Fk+1)·(Fk+2)
. . . . . . . . . . . . . . . . . . . \_______/
. . . . . . . . . . . . . . . . . . . this is Fk+2
. . and we have the RHS of S(k+1).
And the inductive proof is complete.