You can prove that A_{mk} is divisible by A_{m} by induction on k. Then the first term of A_{mk-1}A_{m} + A_{mk}A_{m+1} is clearly divisible by A_{m} and the second term is divisible by the induction hypothesis.
The Fibonacci sequence is given:
A_{1} = A_{2} = 1 , A_{n} = A_{n-1} + A_{n-2}
For m >= 1 and n >= 1 Prove that A_{mn }is divisible by A_{m}.
I have already prepared and proved by induction a Lemma which is:
For m >= 2 and n > = 1. Fibonacci sequence satisfies:
A_{m+n} = A_{m-1}A_{n} + A_{m}A_{n+1}
The consequence of the Lemma allows me to rewrite A_{mn} as:
A_{m(k+1)}= A_{mk+m} = A_{mk-1}A_{m} + A_{mk}A_{m+1}.
So now, I believe I'm suppose to convert that equation to
A_{mk+m} = A_{m}(Some integer).
Been going at it for hours, I'm starting to think I'm approaching it wrong. Thanks in advance!!!
You can prove that A_{mk} is divisible by A_{m} by induction on k. Then the first term of A_{mk-1}A_{m} + A_{mk}A_{m+1} is clearly divisible by A_{m} and the second term is divisible by the induction hypothesis.