Solving a problem in the Fibonacci sequence?

Idea: Let g(n) = f(n+1) -af(n), then we replace n with n+1, that is: g(n+1)=f(n+2) -af(n+1), since there is f(n+2) -a+b)f(n+1) +abf(n) = 0 =>f(n+2) -af(n+1)=bf(n+1)-abf(n)=

b=bg(n) i.e.: g(n+1) = bg(n) This explains why g(n+1) = bg(n) is true

Then, we should think that in order to prove that g(n) is a proportional sequence, we only need to show that g(1) is not equal to 0, and the method is as follows:

Let g(n) = f(n+1) -af(n) in n=1, then g(1) = f(2) -af(1) = 1 - a = b

Therefore, g(n) is a proportional sequence.

g(n+1) = bg(n), that is, g(n+1) g(n)=b, the latter term is b times the previous term, so it is an equal proportional series, and by calculating g(1) = f(2) -af(1) = 1 - a = b, the first term g(1) of g(n) is obtained by calculating g(n), g(1) is the value of f(n+1) -af(n) when n = 1 f(2) -af(1) = 1-a, because, according to a, b, the value calculated earlier, 1-a is equal to b

The Fibonacci sequence is defined as follows:

The Fibonacci sequence, also known as the splitting sequence, was introduced by the mathematician Leonardo Fibonacci (Leonardoda Fibonacci) with the example of rabbit breeding, so it is also called the "rabbit sequence", which refers to such a sequence 、...Mathematically, the Fibonacci sequence is defined recursively as follows: f(0)=0, f(1)=1, f(n)=f(n-1)+f(n-2)(n>=2,n n*) In the fields of modern physics, quasicrystalline structure, chemistry, etc., the Fibonacci sequence has direct applications, for this reason, the American Mathematical Society has published a mathematical journal named "Fibonacci Quarterly Series" since 1963 to publish research results in this area.

The Fibonacci sequence refers to the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368....This sequence starts with term 3, and each term is equal to the sum of the first two terms.

Leonardo Pisano, also known as Fibonacci, Leonardo Bigollo (1175-1250), a medieval Italian mathematician, was the first person in the West to study Fibonacci numbers, and introduced the modern system of place-value notation for written numbers and multipliers to Europe. His book of calculations, written in 1202, contains a lot of Greek, Egyptian, Arab, Indian, and even Chinese mathematics.