\(A=1\cdot2^2+2\cdot3^2+3\cdot4^2+\cdots+98\cdot99^2\)
\(=2^2\left(2-1\right)+3^2\left(3-1\right)+4^2\left(4-1\right)+\cdots+99^2\left(99-1\right)\)
\(=\left(2^3+3^3+\cdots+99^3\right)-\left(2^2+3^2+\cdots+99^2\right)\)
\(=\left(1^3+2^3+\cdots+99^3\right)-\left(1^2+2^2+3^2+\cdots+99^2\right)\)
\(=\left(1+2+3+\cdots+99\right)^2-\frac{99\left(99+1\right)\left(2\cdot99+1\right)}{6}\)
\(=\left(99\cdot\frac{100}{2}\right)^2-\frac{99\cdot100\cdot199}{6}\)
\(=\left(99\cdot50\right)^2-33\cdot50\cdot199=\left(3\cdot33\cdot50\right)^2-33\cdot50\cdot199\)
\(=9\cdot33^2\cdot50^2-33\cdot50\cdot199=33\cdot50\left(9\cdot33\cdot50-199\right)\)
\(=33\cdot50\cdot14651=24174150\)
