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a)Ta có \(\frac{1}{5.8}+\frac{1}{8.11}+\frac{1}{11.14}+...+\frac{1}{x\left(x+3\right)}=\frac{101}{1540}\)
=)\(\frac{3}{5.8}+\frac{3}{8.11}+\frac{3}{11.14}+...+\frac{3}{x\left(x+3\right)}=\frac{303}{1540}\)
=)\(\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+\frac{1}{11}-\frac{1}{14}+...+\frac{1}{x}-\frac{1}{x+3}=\frac{303}{1540}\)
Suy ra \(\frac{1}{5}-\frac{1}{x+3}\)= \(\frac{303}{1540}\)=)\(\frac{1}{x+3}=\frac{1}{305}\)=) \(x+3=305\)=) \(x=302\)
Ta có:
\(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{x.\left(x+1\right):2}=\frac{3984}{1993}\)
\(\Rightarrow\frac{1}{2}.\left(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{x.\left(x+1\right):2}\right)=\frac{1992}{1993}\)
\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x.\left(x+1\right)}=\frac{1992}{1993}\)
\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{1992}{1993}\)
\(1-\frac{1}{x+1}=\frac{1992}{1993}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{1993}\)
\(\Leftrightarrow x+1=1993\)
\(x=1992\)
\(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{x.\left(x+1\right):2}=1\frac{1991}{1993}\)
=> \(\frac{1}{1.\left(1+1\right):2}+\frac{1}{2.\left(2+1\right):2}+\frac{1}{3.\left(3+1\right):2}+\frac{1}{4.\left(4+1\right):2}+...+\frac{1}{x.\left(x+1\right):2}=1\frac{1991}{1993}\)
=> \(\frac{1}{\frac{1.\left(1+1\right)}{2}}+\frac{1}{\frac{2.\left(2+1\right)}{2}}+...+\frac{1}{\frac{x.\left(x+1\right)}{2}}=1\frac{1991}{1993}\)
=> \(\frac{2}{1.2}+\frac{2}{2.3}+...+\frac{2}{x.\left(x+1\right)}=1\frac{1991}{1993}\)
=> \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{x}-\frac{1}{x+1}=1\frac{1991}{1993}\)
=> \(1-\frac{1}{x+1}=1\frac{1991}{1993}\)
=> \(\frac{1}{x+1}=\frac{-1991}{1993}\)
=> -1991.(x + 1) = 1993
=> -1991x - 1991 = 1993
=> -1991x = 3984
=> x = \(-\frac{3984}{1991}\)
A ) \(\frac{1}{3}\left(\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+.....+\frac{1}{x}-\frac{1}{x+3}\right)=\frac{101}{1540}.\)
=\(\frac{1}{3}\left(\frac{1}{5}-\frac{1}{x+3}\right)\)=101/1540
=\(\frac{101}{1540}:\frac{1}{3}=\frac{1}{5}-\frac{1}{x+3}\)
=tới đó bn tự tính nhé
\(\Leftrightarrow\dfrac{1}{3}+\dfrac{1}{6}+...+\dfrac{1}{x\left(x+1\right):2}=\dfrac{1991}{1993}\)
\(\Leftrightarrow\dfrac{2}{6}+\dfrac{2}{12}+...+\dfrac{2}{x\left(x+1\right)}=\dfrac{1991}{1993}\)
\(\Leftrightarrow2\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{x}-\dfrac{1}{x+1}\right)=\dfrac{1991}{1993}\)
\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{x+1}=\dfrac{1991}{3986}\)
=>1/x+1=1/1993
=>x+1=1993
hay x=1992