\(A=1.2^2+2.3^2+...+98.99^2\)
\(=1.2.\left(3-1\right)+2.3.\left(4-1\right)+...+98.99.\left(100-1\right)\)
\(=1.2.3-1.2+2.3.4-2.3+...+98.99.100-98.99\)
\(=\left(1.2.3+2.3.4+...+98.99.100\right)-\left(1.2+2.3+...+98.99\right)\)
\(=\dfrac{98.99.100.101}{4}+\dfrac{98.99.100}{3}\)
\(=24497550+323400\)
\(=24820950\)