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Ta có phần tử \(=\frac{1}{19}+\frac{2}{18}+\frac{3}{17}+...+\frac{18}{2}+\frac{19}{1}\)
\(=\left(\frac{1}{19}+1\right)+\left(\frac{2}{18}+1\right)+...+\left(\frac{18}{2}+1\right)+\left(\frac{19}{1}+1\right)-19\)
\(=\frac{20}{19}+\frac{20}{18}+...+\frac{20}{2}+\frac{20}{1}+\frac{20}{20}-20\)
\(=20.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{19}+\frac{1}{20}\right)\left(1\right)\)
Thay (1) vào P ta được :
\(P=\frac{20.\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{20}\right)}{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{20}}\)
\(=20\)
Bài 1 :
\(S=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2010}-\frac{1}{2011}\)
\(S=\frac{1}{1}-\frac{1}{2011}=\frac{2010}{2011}\)
Bài 2 :
\(S=\frac{1}{10}-\frac{1}{13}+\frac{1}{13}-\frac{1}{16}+\frac{1}{16}-\frac{1}{19}+...+\frac{1}{58}-\frac{1}{61}\)
\(S=\frac{1}{10}-\frac{1}{61}=\frac{51}{610}\)
Bài 3 :
\(3S=\frac{3}{4\times7}+\frac{3}{7\times11}+...+\frac{3}{19\times22}\)
\(3S=\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+...+\frac{1}{19}-\frac{1}{22}\)
\(3S=\frac{1}{4}-\frac{1}{22}\)
\(S=\frac{18}{88}\div3=\frac{6}{88}\)
A = \(1+\frac{1}{\left(2\times3\right):2}+\frac{1}{\left(3\times4\right):2}+....+\frac{1}{\left(19\times20\right):2}\)
A = \(\frac{2}{1\times2}+\frac{2}{2\times3}+\frac{2}{3\times4}+....+\frac{2}{19\times20}\)
A = \(2\times\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{19}-\frac{1}{20}\right)=2\times\left(\frac{1}{1}-\frac{1}{20}\right)=\frac{19}{10}\)
\(a.1\frac{1}{3}+2\frac{1}{2}\)
\(=\frac{4}{3}+\frac{5}{2}\)
\(=\frac{8}{6}+\frac{15}{6}\)
\(=\frac{23}{6}\)
\(b.3\frac{2}{5}-1\frac{1}{7}\)
\(=\frac{17}{5}-\frac{8}{7}\)
\(=\frac{119}{35}-\frac{40}{35}\)
\(=\frac{79}{35}\)
\(c.3\frac{1}{2}.1\frac{1}{7}\)
\(=\frac{7}{2}.\frac{8}{7}\)
\(=4\)
\(d.4\frac{1}{6}:2\frac{1}{3}\)
\(=\frac{25}{6}:\frac{7}{3}\)
\(=\frac{25}{6}.\frac{3}{7}\)
\(=\frac{25}{14}\)
\(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+2015}\)
\(=\frac{1}{\left(1+0\right).2:2}+\frac{1}{\left(1+2\right).2:2}+\frac{1}{\left(1+3\right).3:2}+\frac{1}{\left(1+4\right).4:2}+...+\frac{1}{\left(1+2015\right).2015:2}\)
\(=\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{2015.2016}\)
\(=2.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2015.2016}\right)\)
\(=2.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2015}-\frac{1}{2016}\right)\)
\(=2.\left(1-\frac{1}{2016}\right)\)
\(=2.\frac{2015}{2016}=\frac{2015}{1008}\)