Question

Problema e in atasament! Multumesc!

Answer 1

$\displaystyle Problema~standard~cu~solutie~standard. \\ \\ Se~scrie~relatia~data~astfel: \\ \\ 6f(n)-2f(f(n))=f(n)-3n~\forall~n \in \mathbb{Z}. ~(*)\\ \\ Se~considera~functia~g: \mathbb{Z} \to \mathbb{Z},~g(n)=f(n)-3n. \\ \\ Inlocuind~in~(*)~obtinem~2g(f(n))=g(n)~\forall~n \in \mathbb{Z}. \\ \\ Deci~g(n)=2g(f(n))=2^2(g(f(f(n)))=2^3g(f(f(f(n))))=...= \\ \\ =2^kg(\underbrace{f(f(...(f(n))...))}_{f~apare~de~k~ori}).$

$\displaystyle De~aici~rezulta~ca~g(n)~este~divizibil~cu~2^k~\forall~k \in \mathbb{N}~ceea~ce \\ \\ implica~g(n)=0~\forall~n \in \mathbb{Z}. \\ \\ Deci~g(n)=f(n)-3n=0 ~\forall~n \in \mathbb{Z}. \\ \\ Concluzie:~f(n)=3n~\forall~n \in \mathbb{Z}. \\ \\ Cred~ca~stiu~si~sursa~problemei,~cred...$