\( M=1.2+2.3+3.4+...+216.217\)
\(\Rightarrow3M=1.2.\left(3-0\right)+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+216.217.\left(218-215\right)\)
\(\Rightarrow3M=1.2.3-0.1.2+2.3.4-1.2.3+3.4.5-2.3.4+...+216.217.218-215.216.217\)
\(\Rightarrow3M=-0.1.2+215.216.217\)
\(\Rightarrow M=\dfrac{215.216.217}{3}\)
\(\Rightarrow M=\dfrac{10077480}{3}\)
\(\Rightarrow M=3359160\)