\(\frac{3}{1}+\frac{3}{1+2}+\frac{3}{1+2+3}+...+\frac{3}{1+2+3+...+100}\)
\(=3\times\left(\frac{1}{0+1}+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+...+100}\right)\)
\(=3\times\left(\frac{1}{\left(0+1\right)\times2:2}+\frac{1}{\left(1+2\right)\times2:2}+\frac{1}{\left(1+3\right)\times3:2}+...+\frac{1}{\left(1+100\right)\times100:2}\right)\)
\(=3\times\left(\frac{2}{1\times2}+\frac{2}{2\times3}+\frac{2}{3\times4}+...+\frac{2}{100\times101}\right)\)
\(=3\times2\times\left(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{100\times101}\right)\)
\(=6\times\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{100}-\frac{1}{101}\right)\)
\(=6\times\left(1-\frac{1}{101}\right)\)
\(=6\times\frac{100}{101}\)
\(=\frac{600}{101}\)
Ủng hộ mk nha ^_-
