[tex]n \in N^* \Rightarrow 4n^2\ \textless \ 4n^2+n\ \textless \ 4n^2+4n+1 \Leftrightarrow \\ \\ \Leftrightarrow 4n^2\ \textless \ 4n^2+n\ \textless \ (2n+1)^2 \Rightarrow 2n\ \textless \ \sqrt{4n^2+n}\ \textless \ 2n+1 \Rightarrow \\ \\ \Rightarrow [\sqrt{4n^2+n}]=2n. \\ \\ A=2n^2+2n+1=n^2+(n^2+2n+1)=n^2+(n+1)^2.[/tex]
