S = 1.2 + 2.3 + 3.4 +...+ 99.100
\(3S=1.2.3+2.3.3+3.4.3+...+99\cdot100\cdot3\)
\(3S=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+99\cdot100.\left(101-98\right)\)\(3S=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+4.5.6-3.4.5+...+99.100.101-98.99.100\)\(3S=99.100.101\)
\(S=\dfrac{99.100.101}{3}\)
\(S=333300\)