\(A=1\left(2+2\right)+2\left(2+3\right)+3\left(2+4\right)+.....+\left(n-1\right)\left(2+n\right)\)
\(\Leftrightarrow A=1.2+1.2+2.3+2.2+3.4+2.3+....+\left(n-1\right)n+2\left(n-1\right)\)
\(\Leftrightarrow A=\left(1.2+2.3+.....+\left(n-1\right)n\right)+2\left(1+2+3+....+\left(n-1\right)\right)\)
Giả sử A=B+C
Với \(\begin{cases}B=1.2+2.3+.....+\left(n-1\right)n\\C=2\left[1+2+....+\left(n-1\right)\right]\end{cases}\)
Ta có
\(3B=1.2.\left(3-0\right)+2.3.\left(4-1\right)+......+\left(n-1\right)n\left[\left(n+1\right)-\left(n-2\right)\right]\)
\(\Rightarrow3B=1.2.3-0.1.2+2.3.4-1.2.3+.....+\left(n-1\right)n\left(n+1\right)-\left(n-2\right)\left(n-1\right)n\)
\(\Rightarrow B=\frac{\left(n-1\right)n\left(n+1\right)}{3}\)
Mặt khác
\(C=2\left[1+2+....+\left(n-1\right)\right]\)
\(\Rightarrow C=2.\frac{\left[\left(n-1\right)+1\right]n}{2}=n^2\)
\(\Rightarrow A=\frac{\left(n-1\right)n\left(n+1\right)}{3}+n^2\)
Vậy \(A=\frac{\left(n-1\right)n\left(n+1\right)}{3}+n^2\)