The Fibonacci sequence is defined as f_1=f_2=1,f_(n+2)=f_(n+1)+f_n. Assume that (f_n)/(f_(n-1))<a/b<(f_(n+1))/(f_n) (a and b are positive integers with no common prime factors). Show that b>=f_(n+1).
Roderick Bradley
Answered question
2022-08-11
Show that The Fibonacci sequence is defined as . Assume that ( and are positive integers with no common prime factors). Show that
Answer & Explanation
Ayla Coffey
Beginner2022-08-12Added 8 answers
Remark: are convergents of the continued fraction of the golden ratio. There is a theorem that any convergent is nearer to the number (whose CF we are looking at) than any other rational with a smaller denominator. That or something similar can potentially be used here. Anyway, here is a simple proof that First we show that Subtract from the given inequality to get
We used the identity (Cassini's formula) to simplify the right hand side. i.e
And so
Now is a positive integer and so is and so Now we show that Now if , then we will have So assume that
Now all solutions of
are given by
where is an integer param, and are some initial solution. For us (again using Cassini's)
Thus the solutions are
Since we want
we need in which case
Makayla Eaton
Beginner2022-08-13Added 6 answers
Suppose are non-negative integers and are positive integers with and Then
Since these are all integers we have
Multiplying the left inequality above by and multiplying the right inequality above by we have