\(\frac12+\frac{1}{12}+\frac{1}{30}+\cdots+\frac{1}{9900}\)

\(=\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\cdots+\frac{1}{99\cdot100}\)

\(=1-\frac12+\frac13-\frac14+\cdots+\frac{1}{99}-\frac{1}{100}\)

\(=\left(1+\frac12+\cdots+\frac{1}{99}+\frac{1}{100}\right)-2\left(\frac12+\frac14+\cdots+\frac{1}{100}\right)\)
\(=1+\frac12+\cdots+\frac{1}{99}+\frac{1}{100}-1-\frac12-\cdots-\frac{1}{50}\)

\(=\frac{1}{51}+\frac{1}{52}+\cdots+\frac{1}{100}\)

Ta có: \(A=\left(\frac12+\frac{1}{12}+\frac{1}{30}+\cdots+\frac{1}{9900}\right):\left(\frac{-6}{51}-\frac{6}{52}-\cdots-\frac{6}{100}\right)\)

\(=\frac{\frac{1}{51}+\frac{1}{52}+\cdots+100}{-6\left(\frac{1}{51}+\frac{1}{52}+\cdots+\frac{1}{100}\right)}=\frac{-1}{6}\)

(2 x X - 1)mũ 3 = (2 x X - 2)mũ 2

Ta có: \(\frac{1}{51}>\frac{1}{75};\frac{1}{52}>\frac{1}{75};\ldots;\frac{1}{74}>\frac{1}{75};\frac{1}{75}=\frac{1}{75}\)

Do đó: \(\frac{1}{51}+\frac{1}{52}+\cdots+\frac{1}{75}>\frac{1}{75}+\frac{1}{75}+\cdots+\frac{1}{75}=\frac{25}{75}=\frac13\) (1)

Ta có: \(\frac{1}{76}>\frac{1}{100};\frac{1}{77}>\frac{1}{100};\ldots;\frac{1}{99}>\frac{1}{100};\frac{1}{100}=\frac{1}{100}\)

Do đó: \(\frac{1}{76}+\frac{1}{77}+\cdots+\frac{1}{100}>\frac{1}{100}+\frac{1}{100}+\cdots+\frac{1}{100}=\frac{25}{100}=\frac14\) (2)

Từ (1),(2) ta có: \(\frac{1}{51}+\frac{1}{52}+\cdots+\frac{1}{75}+\frac{1}{76}+\frac{1}{77}+\cdots+\frac{1}{100}>\frac13+\frac14\)

=>\(S>\frac13+\frac14=\frac{7}{12}\) (3)

Ta có: \(\frac{1}{51}<\frac{1}{50};\frac{1}{52}<\frac{1}{50};\ldots;\frac{1}{75}<\frac{1}{50}\)

Do đó: \(\frac{1}{51}+\frac{1}{52}+\cdots+\frac{1}{75}<\frac{1}{50}+\frac{1}{50}+\cdots+\frac{1}{50}=\frac{25}{50}=\frac12\) (4)

Ta có: \(\frac{1}{76}<\frac{1}{75};\frac{1}{77}<\frac{1}{75};\ldots;\frac{1}{100}<\frac{1}{75}\)

Do đó: \(\frac{1}{76}+\frac{1}{77}+\cdots+\frac{1}{100}<\frac{1}{75}+\frac{1}{75}+\cdots+\frac{1}{75}=\frac{25}{75}=\frac13\) (5)

Từ (4),(5) suy ra \(\frac{1}{51}+\frac{1}{52}+\cdots+\frac{1}{75}+\frac{1}{76}+\frac{1}{77}+\cdots+\frac{1}{100}<\frac12+\frac13\)

=>\(S<\frac56\) (6)

Từ (3),(6) suy ra 7/12<S<5/6

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