The general formula of Fibonacci sequence is fn = a n+b n (n ≥ 1), where a and b satisfy the equations a+b = 0 and a 2+b 2 =1. By solving this set of equations, we can get a =1√ 5 and b =-1√ 5. Therefore, the general term formula of Fibonacci series can be further simplified as: fn = (1√ 5) n-(-1√ 5) n This is the derivation process of the general term formula of Fibonacci series.

Extended data

Fibonacci series, also known as the golden section series, was introduced by mathematician Leonardo Fibonacci taking raising rabbits as an example, so it is also called "rabbit series", and its numerical values are: 1, 1, 2, 3, 5, 8, 13. This series is defined by the following recursive methods: F(0)= 1, f (1) = 1, f (n) = f (n- 1)+f (n-2) (n ≥ 2, n.

origin

Fibonacci was the first influential mathematician in the history of mathematics. In his early years, he studied arithmetic with Arabs in North Africa with his father, and then traveled to Mediterranean countries. After returning to Italy, he wrote a calculation book and translated it into an abacus book. This masterpiece mainly collects mathematical problems in ancient China, Indian and Greek, involving integer and fractional algorithms, open methods, quadratic and cubic equations and indefinite equations.

In particular, the revised edition of Computational Classics in 1228 contains the following "rabbit problem": if each pair of rabbits (one male and one female) can give birth to one pair of rabbits every month (also a male and one female, the same below), each pair of rabbits has no reproductive ability in the first month, and can give birth to one pair of rabbits every month after the second month. Assuming these rabbits are not dead, from the first month,

The explanation is: one month: only one pair of rabbits; The second month: still only a pair of rabbits; The third month: The rabbits gave birth to a pair of rabbits, * * * with 1+ 1=2 pairs of rabbits. The fourth month: the first pair of rabbits gave birth to another pair of rabbits, * * * with 2+ 1=3 pairs of rabbits.

Then the logarithms of rabbits from the first month to the twelfth month are: 1, 1, 2, 3, 5, 8, 13, 2 1, 34, 55, 89, 144, ...