Let $\mathbb N=\{1,2,3,…\}$. Determine if there exists a strictly increasing function $f:\mathbb N→\mathbb N$ such that
$f(1)=2$
$f(f(n))=f(n)+n$ for all $n$.
Recently I posted the following solution I came up with
We can see
$ f(1)=2$
$f(2)=3$
$f(3)=5 $
Basically if $a_n$ is the $(n+2)$th term in the Fibonacci sequence starting from $1,1,2,3,5...$
then I claim
$f(a_n)=a_{n+1}$
This can easily be proved by induction as follows
Assume $f(a_n)=a_{n+1}$.
Then $f(a_{n+1})=f(a_n)+a_n=a_{n+2}$.
Now clearly $$a_{n+1}−a_n<a_{n+2}−a_{n+1},$$ so all numbers between $a_n$ and $a_{n+1}$ can be assigned any arbitrary value between $a_{n+2}$ and $a_{n+1}$ in increasing order and hence many such functions may exist.
But as it was pointed out by a couple of users that in the last last step assigning arbitrary values may not fulfill condition 2.
Hence can someone tell me how to adress that problem without actually finding a function satisfying the functional equation as we only need to tell that such a function exists?