[tex]A=(-1)^{n^2+n}\cdot 1+1\cdot (-1)^{n^3+2n+3}-(-1)^{n^2+4n-5}=\\
(-1)^{n(n+1)}+(-1)^{n^3+2n+3}-(-1)^{n^2+4n-5}\\
1) n\ par\\
n(n+1)\ par; (-1)^{n(n+1)}=1\\
n^3+2n+3 \ impar; (-1)^{n^3+2n+3}=-1\\
n^2+4n-5\ impar; (-1)^{n^2+4n-5}=-1\\
A=1-1-(-1)=1-1+1=1\\
2) n \ impar\\
n(n+1)\ par; (-1); (-1)^{n(n+1)}=1\\
n^3+2n+3\ par; (-1)^{n^3+2n+3}=1\\
n^2+4n-5 \ par; (-1)^{n^2+4n-5}=1\\
A=1+1-1=1\\
deci \ A=1 [/tex]
(∀) [tex]n\in N[/tex]
