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Gọi A=\(\frac{1}{5.9}+\frac{1}{9.13}+....+\frac{1}{41.45}\)
=>4A=\(\frac{4}{5.9}+\frac{4}{9.13}+....+\frac{4}{41.45}\)
4A=\(\frac{9-5}{5.9}+\frac{13-9}{9.13}+....+\frac{45-41}{41.45}\)
4A\(=\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+....+\frac{1}{41}-\frac{1}{45}\)
\(4A=\frac{1}{5}-\frac{1}{45}\)
\(4A=\frac{8}{45}\)
\(=>A=\frac{2}{45}\)
\(\frac{1}{1.3.7}+\frac{1}{3.7.9}+\frac{1}{7.9.13}+\frac{1}{9.13.15}+\frac{1}{13.15.19}\)
\(=\frac{1}{2}\left(\frac{1}{1.3}-\frac{1}{3.7}+\frac{1}{3.7}-\frac{1}{7.9}+...+\frac{1}{13.15}-\frac{1}{15.19}\right)\)
\(=\frac{1}{2}\left(\frac{1}{1.3}-\frac{1}{15.19}\right)=\frac{47}{285}\)
\(\frac{1}{1.3.7}=\frac{1}{6}\left(\frac{1}{1.3}-\frac{1}{3.7}\right)\)
\(\frac{1}{3.7.9}=\frac{1}{6}\left(\frac{1}{3.7}-\frac{1}{7.9}\right)\)
....
\(\frac{1}{13.15.19}=\frac{1}{6}\left(\frac{1}{13.15}-\frac{1}{15.19}\right)\)
Cộng các vế với nhau ta được
\(\frac{1}{1.3.7}+\frac{1}{3.7.9}+...+\frac{1}{13.15.19}=\frac{1}{6}\left(\frac{1}{1.3}-\frac{1}{15.19}\right)=\frac{37}{3.15.19}\)
\(C=\frac{2}{1\cdot5}+\frac{2}{5\cdot9}+\frac{2}{9\cdot13}+\frac{2}{13\cdot17}+\frac{2}{17\cdot21}\)
\(C=\frac{2}{4}.\left(\frac{4}{1.5}+\frac{4}{5.9}+\frac{4}{9.13}+\frac{4}{13.17}+\frac{4}{17.21}\right)\)
\(=\frac{1}{2}.\left(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+....+\frac{1}{17}-\frac{1}{21}\right)\)
\(=\frac{1}{2}.\left(1-\frac{1}{21}\right)\)
\(=\frac{1}{2}.\frac{20}{21}\)
\(=\frac{10}{21}\)
gạch tất cả số 5, 9, 13
là bằng 4.x/1 + 4.x/17
rồi gợi ý thế thôi nhé
\(\frac{4.x}{1.5}+\frac{4.x}{5.9}+\frac{4.x}{9.13}+\frac{4.x}{13.17}=16\)
\(x.\left(\frac{4}{1.5}+\frac{4}{5.9}+\frac{4}{9.13}+\frac{4}{13.17}\right)=16\)
\(x.\left(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+\frac{1}{13}-\frac{1}{17}\right)=16\)
\(x.\left(1-\frac{1}{17}\right)=16\)
\(x.\frac{16}{17}=16\Rightarrow x=16:\frac{16}{17}=16.\frac{17}{16}\)
\(\Rightarrow x=17\)
\(A=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2014.2015}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-...+\frac{1}{2014}-\frac{1}{2015}=1-\frac{1}{2015}=\frac{2014}{2015}\)
\(B=\frac{2}{11.13}+\frac{2}{13.15}+...+\frac{2}{97.99}=\frac{2}{11}-\frac{2}{13}+...+\frac{2}{97}-\frac{2}{99}=\frac{2}{11}-\frac{2}{99}=\frac{16}{99}\)
C=(1x2)+(2x3)+...+(99x100)
3xC=1x2x3+2x3x3+...+3x99x100
3xC=1x2x3-0x1x2+2x3x4-1x2x3+...+99x100x101-98x99x100
3xC=99x100x101
3xC=999900
C=333300
gọi giá trị biểu thức là A
A = 1/5.9+ 1/9.13+.......+ 1/41.45
A.4 = 4. ( 1/5.9+ 1/9.13+.......+ 1/41.45 )
= 4/5.9+ 4/9.13+.......+ 4/41.45
=1/5- 1/9+ 1/9-1/13 +.....+1/41- 1/45
=1/5-1/45
=8/45
A = 8/45 : 4
=2/45