Question

aratati ca numarul x=5^2n+1 +31x25^n,este patrat perfect ,oricare ar fi nEN

Answer 1

$\displaystyle x=5^{2n+1}+31 \cdot 25^n= \\ \\ =5 \cdot 5^{2n}+31 \cdot 25^n= \\ \\ =5 \cdot 25^n+31 \cdot 25^n= \\ \\ =25^n(5+31)= \\ \\ =25^n \cdot 36= \\ \\ =(5^n)^2 \cdot 6^2= \\ \\ =(6 \cdot 5^n)^2 \longleftarrow patrat~perfect$