Question

$Se~considera~x~\in~R~cu~proprietatea~x+ \frac{1}{x} \in~Z. \\ Demonstrati~ca: \\ a) x^{n} + \frac{1}{ x^{n} } ~\in~Z,~x~\in~ N^{*}$

Answer 1

Fie x+1/x=k∈Z =>
(x+1/x)²=k²∈Z
Presupunem propozizitia Pn =x^n+1/x^n∈Z  adevarata.Verificam  daca  si  Pn+1=(x^(n+1)+1/x^(n+1)∈Z
Pn=(x^n+1/x^n∈Z (A
Pn+1=(x^(n+1)+1/x^(n+1)  (B
(x^n+1/x^n)*(x+1/x)∈Z  pt  ca  ambii   factori apartin  lui Z
x^n*x+x/x^n+x^n/x+1/x^(n+1)=[x^(n+1)+1/x^(n+1)]+[1/x^(n-1)+x^(n-1)]∈Z
Dar  conf  A si  paranteza   2-a  e   nr  intreg  pt  ca  n-1<n
Deci  si  prima  paranteza e  un  nr  intreg
A