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19 nha !!!!!!!!!! tick mình đi làm ơn để mình đủ 20 không mình bị phạt đấy
\(S=\frac{-1}{2}-\frac{1}{3}-\frac{1}{4}-...-\frac{1}{20}+\frac{3}{2}+\frac{4}{3}+\frac{5}{4}+...+\frac{21}{20}\)
\(S=\left(\frac{3}{2}-\frac{1}{2}\right)+ \left(\frac{4}{3}-\frac{1}{3}\right)+\left(\frac{5}{4}-\frac{1}{4}\right)+...+\left(\frac{21}{20}-\frac{1}{20}\right)\)
\(S=1+1+1...+1\)
\(S=1.20=20\)
\(S=-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}-...-\frac{1}{20}+\frac{1}{2}+\frac{2}{2}+\frac{1}{3}+\frac{3}{3}+...+\frac{1}{20}+\frac{20}{20}\)
\(S=\left(-\frac{1}{2}+\frac{1}{2}\right)-\left(\frac{1}{3}-\frac{1}{3}\right)-...-\left(\frac{1}{20}-\frac{1}{20}\right)+\left(1+1+...+1\right)\)
\(=0+0+...+0+1\cdot19=19\)
\(\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{n+1}\right)\)
\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{n}{n+1}\)
\(=\frac{1}{n+1}\)
\(1+\frac{1}{2}.\left(1+2\right)+\frac{1}{3}.\left(1+2+3\right)...+\frac{1}{20}.\left(1+2+3+...+20\right)\)
\(=1+\frac{1}{2}.2.3:2+\frac{1}{3}.3.4:2+\frac{1}{4}.4.5:2+...+\frac{1}{20}.20.21:2\)
\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+\frac{5}{2}+...+\frac{21}{2}\)
\(=\frac{2+3+4+5+...+21}{2}=115\)
Huong dẫn: \(1+2+...+n=\frac{n\left(n+1\right)}{2}\) ( n\(\in\)N*) áp dụng vào từng cái ngoặc
\(S=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+.......+\frac{1}{20}\left(1+2+3+......+20\right)\)
\(=1+\frac{1}{2}.\frac{2.3}{2}+\frac{1}{3}.\frac{3.4}{2}+......+\frac{1}{20}.\frac{20.21}{2}\)
\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+......+\frac{21}{2}=\frac{2+3+4+.....+21}{2}=\frac{20.23}{2}=230\)