Question

aratati ca $\sqrt{ n^{2}+n+1 n^{2}+n+3+1 }$ apartine N pentru orice n apartine N

Answer 1

$A= \sqrt{(n^2+n+1)(n^2+n+3)+1}~. \\ \\ Notam~n^2+n+1=a. \\ \\ A= \sqrt{a(a+2)+1}= \sqrt{a^2+2a+1}= \sqrt{(a+1)^2}=|a+1|= \\ \\ =|n^2+n +2|=n^2+n+2 \in N.$