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\(B=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+\frac{1}{4}\left(1+2+3+4\right)+.......+\frac{1}{20}\left(1+2+3+....+20\right)\)
\(B=1+\left(\frac{1}{2}+1\right)+2+\left(\frac{1}{2}+2\right)+3+\left(\frac{1}{2}+3\right)+.....+10+\left(\frac{1}{2}+10\right)\)(chỗ này là nhân phân phối vô đấy!)
\(B=\left(1+2+3+....+10\right)+\left(1+2+3+...+10\right)+\left(\frac{1}{2}.10\right)\)
\(B=55+55+5=115\)
\(1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+...+\frac{1}{20}\left(1+2+3+...+20\right)=1+\frac{1}{2}.\frac{2.3}{2}+\frac{1}{3}.\frac{3.4}{2}+...+\frac{1}{20}.\frac{20.21}{2}=1+\frac{3}{2}+\frac{4}{2}+...+\frac{21}{2}=1+\frac{24.19}{2}=229\)
\(\left(1-\frac{1}{2}\right).\left(1-\frac{1}{3}\right).\left(1-\frac{1}{4}\right)...........\left(1-\frac{1}{19}\right).\left(1-\frac{1}{20}\right)\)
\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.............\frac{18}{19}.\frac{19}{20}\)
\(=\frac{1.2.3...........18.19}{2.3.4...................19.20}=\frac{1.\left(2.3.4..........18.19\right)}{\left(2.3.............19\right).20}=\frac{1}{20}\)
\(1+\frac{1}{2}.\left(1+2\right)+\)\(\frac{1}{3}.\left(1+2+3\right)+\frac{1}{4}.\left(1+2+3+4\right)+...+\frac{1}{16}.\left(1+2+3+...+16\right)\)
=\(\frac{2}{2}+\frac{3}{2}+\frac{6}{3}+...+\frac{136}{16}\)
=\(\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+...+\frac{17}{2}\)
=\(\frac{2+3+4+5+6+...+17}{2}\)=\(\frac{152}{2}=76\)
Ta có:
\(1+\frac{1}{2}\left(1+2\right)+..........+\frac{1}{20}\left(1+2+3+.......+20\right)\)
\(=1+\frac{1}{2}\left(\frac{3.2}{2}\right)+\frac{1}{3}\left(\frac{4.3}{2}\right)+........+\frac{1}{20}\left(\frac{21.20}{2}\right)\)
\(=1+\frac{3}{2}+\frac{4}{2}+..........+\frac{21}{2}=\frac{2+3+4+........+21}{2}\)
\(=\frac{\frac{23.20}{2}}{2}=\frac{23.10}{2}=115\)