John R. Simmons Proof by induction
Let P(n) be the statement "f1+f2+f3+...fn=f(n+2) - 1
1.) Since f1=f(1=2) - 1 = f3 - 1 = 2 - 1 = 1
P(1) is true.
2.) Suppose that p(k) is true for a positive integer k.
Then f1+f2+...+fk = f(k+2) - 1
Therefore, f1+f2+...fk+f(k+1) = f(k+2) - 1 + f(k+1)
= f(k+1 + f(k+2) - 1 = f(k+3) - 1
because f(k+1) + f(k+2) = f(k+3) by definition of Fibonacci numbers
f(k+3) = f(k+3) - 1 + f(k+3) -2. QED.
