\(S=\dfrac{1}{1\cdot3}+\dfrac{1}{2\cdot4}+\dfrac{1}{3\cdot5}+\dfrac{1}{4\cdot6}+...+\dfrac{1}{7\cdot9}+\dfrac{1}{6\cdot8}\)
\(=\left(\dfrac{1}{1\cdot3}+\dfrac{1}{3\cdot5}+...+\dfrac{1}{7\cdot9}\right)+\left(\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+\dfrac{1}{6\cdot8}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{7\cdot9}\right)+\dfrac{1}{2}\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+\dfrac{2}{6\cdot8}\right)\)
\(=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{7}-\dfrac{1}{9}\right)+\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}\right)\)
\(=\dfrac{1}{2}\left(1-\dfrac{1}{9}\right)+\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{8}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{9}{9}-\dfrac{1}{9}\right)+\dfrac{1}{2}\left(\dfrac{4}{8}-\dfrac{1}{8}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{8}{9}+\dfrac{1}{2}\cdot\dfrac{3}{8}\)
\(=\dfrac{1}{2}\left(\dfrac{8}{9}+\dfrac{3}{8}\right)\)
\(=\dfrac{1}{2}\left(\dfrac{64}{72}+\dfrac{27}{72}\right)\)
\(=\dfrac{1}{2}\cdot\dfrac{91}{72}\)
\(=\dfrac{91}{144}\)
S=\(\dfrac{1}{1.3}+\dfrac{1}{2.4}+...+\dfrac{1}{6.8}\)
S=\(\dfrac{1}{2}.\left(\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{2}+...+\dfrac{1}{6}-\dfrac{1}{8}\right)\)
S=\(\dfrac{1}{2}.\left(\dfrac{1}{1}-\dfrac{1}{8}\right)\)
S=\(\dfrac{1}{2}.\left(\dfrac{8-1}{8}\right)\)
S=\(\dfrac{1}{2}.\dfrac{7}{8}\)
S=\(\dfrac{7}{16}\)
S=11⋅3+12⋅4+13⋅5+14⋅6+...+17⋅9+16⋅8S=11⋅3+12⋅4+13⋅5+14⋅6+...+17⋅9+16⋅8
=(11⋅3+13⋅5+...+17⋅9)+(12⋅4+14⋅6+16⋅8)=(11⋅3+13⋅5+...+17⋅9)+(12⋅4+14⋅6+16⋅8)
=12(21⋅3+23⋅5+...+27⋅9)+12(22⋅4+24⋅6+26⋅8)=12(21⋅3+23⋅5+...+27⋅9)+12(22⋅4+24⋅6+26⋅8)
=12(1−13+13−15+...+17−19)+12(12−14+14−16+16−18)=12(1−13+13−15+...+17−19)+12(12−14+14−16+16−18)
=12(1−19)+12(12−18)=12(1−19)+12(12−18)
=12(99−19)+12(48−18)=12(99−19)+12(48−18)
=12⋅89+12⋅38=12⋅89+12⋅38
=12(89+38)=12(89+38)
=12(6472+2772)=12(6472+2772)
=12⋅9172=12⋅9172
=91144
