Fibonacci numbers

Question

F_N denotes the Nth fibonacci number.

 

Given the following fact: (F₁ + F₂ + F₃ + F₄ + ..... + F_N) + 1 = F_N+2

 

How would I verify this fact?

 

For example, verify this fact if N = 11.

Answer 1

Christopher,

The Fibonacci sequence starts as follows:

 

1 1 2 3 5 8 13......

 

Aside from the first two ones, every term is the sum of the previous two terms. So to verify the stated relationship, just continue the list to the 13th term.  Add the first 11 terms plus one and see if that matches the 13th term.

Answer 2

Are you trying to verify the formula itself holds for all N?

 

Try using a proof by induction:

 

Let P(N) be the "fact" you stated in the question.

 

Step 1: Prove the formula is true for N=1             (basis step)

 

Step 2: Assume true for N=k                               (inductive hypothesis)

 

Step 3: Prove it is true for N=k+1                        (inductive proof)

 

 

Any questions? Comment back.