Fibonacci

The Fibonacci Numbers

We define the

n

-th fibonacci number as follows

Sum Equals Jump 2 minus One

For every

n \in ℕ_{0}

we have

F_{0} + F_{1} + \dots + F_{n} = F_{n + 2} - 1

We prove it by induction, for the base case

F (0 + 2) - 1 = (F_{1} + F_{0}) - 1 = 1 + 0 - 1 = 0 = F_{0}

as needed. Next assume it holds true for

k \in ℕ_{0}

and then we want to prove that

F_{k + 3} - 1 = F_{0} + \dots + F_{k + 1}

, we can handle the sum with the inductive hypothesis, so that

F_{0} + \dots + F_{k} + F_{k + 1} = (F_{k + 2} - 1) + F_{k + 1} = F_{k + 3} - 1

as needed.