Consideration on Fibonacci Numbers

By Mihai Rosu 

Let                         ,                                                     (1)

be the recurrent sequence.   Suppose that , , , , ,                          

Writing the terms of this sequence, we get

                 0, 1, 1, 2, 3, 5, 8, …                                                                        (2)         

called the Fibonacci sequence and F is called the n-th Fibonacci numbers.

Fibonacci was one of the greatest European mathematician of the middle ages, his full name was Leonardo of Pisa, or Leonardo Pisano in Italian, he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD.

The Lucas numbers  are defined by the equations  ,     (3)

and satisfy the same recurrence where, the first few are

 1, 3, 4, 7, 11, 18, 29, 47, 76,123, ...

The French mathematician, Edouard Lucas (1842-1891), was the first who gave the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, .. the name the Fibonacci Numbers.

Assuming that the sequence F_n has the form  , where  is a real parameter.

Substituting  in (1)

                                                                                                (4)

or equivalent                

But     ("n  N^*), the last equality becomes

This is a quadratic equation with res pect to the real parameter l having the roots

                             and                                                (5)

Thus the sequences  ,

verify the equality (1). So we conclude that the equation (1) can have more solutions. In general there are an infinite of the sequences verifying (1). Easy to observe that (1) has the form                                                       (6)

where c₁, c₂ –are fixed real numbers, verifying (1), too. Also you can prove that any sequence verifying (1) has the form (6).

For n = 0 and n = 1 in (6), we get the linear system

having the solutions 

Finally, the general term of the Fibonacci sequence has the form

                          N.                  (7)

Some Proprieties of the Fibonacci sequence.

1.                                                                  (8)

Proof:           

                      …………….

Summing all equalities we get  ,

but    so (8) is  shown.

2.       .

3.      .

      2. and 3.  can be proven in a similar manner.

4.                                                                (9)

Proof: It is easy to observe that ,  .

From this we have successively the equalities:       

                      ………………….

Summing all these equalities we get (9).

5.        Prove that                                                (10)

where F_n is the n-th term of the Fibonacci sequences.

Proof: Using (1) and (9) then (10) is easy to be proven.

But my goal is to prove (10) using the mathematical induction. I’ll proceed by mathematical induction with respect .

For m = 1, the equality (10) becomes , this being evident, then

(10) is true for m=1. (For example when m = 2 the formula (10) is true

).

Assuming that (10) is true for m = k and m = k + 1.

I’ll prove that (10) is true for m = k + 2, too.

Therefore, being true the equalities        

                                                                ,

Summing both equalities we get            ,

in fact this is (10) for m = k + 2.

6. Prove that   (sometimes called double angle formula),

Proof:  If in  (10)  m = n   we get  ,

7. Prove that   ,    n>1                   (11)

Proof: Again I’ll proceed for (11) by mathematical induction.

For n = 2 then (11) becomes  , which is true. Thus (11) is true for n=2.

Assuming that (11) is true for n, I’ll prove that it is true for n + 1, too.

So   is true. Adding in both sides in the last equality the number

  I get  or

.

Next using (1) I have  and finally  .

Hence condition (11) is true for n + 1.