How can I prove by induction that $3 ∣ f _{4n}$ is true?

Question

This is related to the fibonaci numbers. I understand the way a proof by induction works, and these are 2 statements that are given:

$f_1=1$

$f_2=1$

$f_n = f_{n-1} + f_{n-2}$,

as long as $f\ge3$ (greater than or equal to 3).

I have been stuck on this for a while now, i cant get past the proof with n=1.

Use $F_{n+k}+(-1)^{k}F_{n-k}=L_{k}F_{n}$ with $k=4$, mentioned in https://en.wikipedia.org/wiki/Lucas_number#Relationship_to_Fibonacci_numbers. Prove it by induction. See also https://math.stackexchange.com/questions/3370773/proof-of-the-identity-f-n1k-fracf-2kf-kf-nk-1k1f-n-1k. — lhf, Nov 22 '22 at 18:41
Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. — Another User, Nov 22 '22 at 18:41
You do not tell what you want to prove? you just give the definition of the Fibonacci numbers — trula, Nov 22 '22 at 18:41
If $g_0=a, g_1=b$ with $a,b$ integers and $d\mid a,$ and $g_{n+1}=g_n+g_{n-1},$ then prove by induction that, for each $n,$ $$g_n\equiv bf_n\pmod d,$$ where $f_n$ is the usual Fibonacci sequence. — Thomas Andrews, Nov 22 '22 at 18:44
Then if $a=f_{4n},b=f_{4n+1}$ if $3\mid a,$ then $$f_{4n+4}=g_4\equiv bf_4=3b\equiv 0\pmod 3$$ — Thomas Andrews, Nov 22 '22 at 18:47
Does this answer your question? Let $(F_n)_{n \in \mathbb{N}}$ be a Fibonacci sequence , then $(4 \mid n \Leftrightarrow 3 \mid F_n)$ for all $n \in \mathbb{N}$ - found using an Approach0 search. Another basic duplicate is Is it true that the Fibonacci sequence has the remainders when divided by 3 repeating?. Also, using $f_{4}=3$, there are quite a ... — John Omielan, Nov 22 '22 at 19:19
(cont.) few posts regarding more general Fibonacci number divisibility properties which could be used, e.g., Proving that $n|m\implies f_n|f_m$, Fibonacci divisibilty properties $ F_n\mid F_{kn},,$ $, \gcd(F_n,F_m) = F_{\gcd(n,m)}$, etc. — John Omielan, Nov 22 '22 at 19:23

How can I prove by induction that $3 ∣ f _{4n}$ is true?

0 Answers0