** Mathematics Magazine
for Grades 1-12 **

**10/2003**

## Consideration on Fibonacci NumbersBy 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 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 Substituting in (1) (4) or equivalent But
(" 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 For
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-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. Ill proceed by mathematical induction with respect . For (10) is true for
m=1. (For example when ). Assuming that (10)
is true for Ill prove that
(10) is true for Therefore, being true the equalities , Summing both
equalities we get
, in fact this is (10)
for 6. Prove that
(sometimes called Proof:
If in (10)
7. Prove that
, Proof: Again Ill proceed for (11) by mathematical induction. For Assuming that (11)
is true for 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 |