Khẳng định nào sau đây đúng nhất:

Nếu a ∈ ℤ thì

Ta có:

$\frac{9}{20}=\frac{9}{4.5}=\frac{5+4}{4.5}=\frac{5}{4.5}+\frac{4}{4.5}=\frac{1}{4}+\frac{1}{5}$

$\frac{11}{30}=\frac{11}{5.6}=\frac{6+5}{5.6}=\frac{6}{5.6}+\frac{5}{5.6}=\frac{1}{5}+\frac{1}{6}$

$\frac{13}{42}=\frac{13}{6.7}=\frac{7+6}{6.7}=\frac{7}{6.7}+\frac{6}{6.7}=\frac{1}{6}+\frac{1}{7}$

…

$\frac{197}{9702}=\frac{197}{98.99}=\frac{99+98}{98.99}=\frac{99}{98.99}+\frac{98}{98.99}=\frac{1}{98}+\frac{1}{99}$

$\frac{199}{9900}=\frac{199}{99.100}=\frac{100+99}{99.100}=\frac{100}{99.100}+\frac{99}{99.100}=\frac{1}{99}+\frac{1}{100}$

Do đó $H=\frac{38}{50}+\frac{9}{20}-\frac{11}{30}+\frac{13}{42}-\frac{15}{56}+\frac{17}{72}-\mathrm{...}+\frac{197}{9702}-\frac{199}{9900}$

$$\begin{gathered} =\frac{38}{50}+\left(\frac{1}{4}+\frac{1}{5}\right)-\left(\frac{1}{5}+\frac{1}{6}\right)+\left(\frac{1}{6}+\frac{1}{7}\right)-\left(\frac{1}{7}+\frac{1}{8}\right)++\left(\frac{{\displaystyle 1}}{{\displaystyle 8}}+\frac{{\displaystyle 1}}{{\displaystyle 9}}\right)-\mathrm{...}+\left(\frac{{\displaystyle 1}}{{\displaystyle 98}}+\frac{{\displaystyle 1}}{{\displaystyle 99}}\right)-\left(\frac{{\displaystyle 1}}{{\displaystyle 99}}+\frac{{\displaystyle 1}}{{\displaystyle 100}}\right) \\ =\frac{{\displaystyle 38}}{{\displaystyle 50}}+\frac{{\displaystyle 1}}{{\displaystyle 4}}+\frac{{\displaystyle 1}}{{\displaystyle 5}}-\frac{{\displaystyle 1}}{{\displaystyle 5}}-\frac{{\displaystyle 1}}{{\displaystyle 6}}+\frac{{\displaystyle 1}}{{\displaystyle 6}}+\frac{{\displaystyle 1}}{{\displaystyle 7}}-\frac{{\displaystyle 1}}{{\displaystyle 7}}-\frac{{\displaystyle 1}}{{\displaystyle 8}}+\frac{{\displaystyle 1}}{{\displaystyle 8}}+\frac{{\displaystyle 1}}{{\displaystyle 9}} \\ =\frac{{\displaystyle 38}}{{\displaystyle 50}}+\frac{{\displaystyle 1}}{{\displaystyle 4}}+\left(\frac{{\displaystyle 1}}{{\displaystyle 5}}-\frac{{\displaystyle 1}}{{\displaystyle 5}}\right)+\left(-\frac{{\displaystyle 1}}{{\displaystyle 6}}+\frac{{\displaystyle 1}}{{\displaystyle 6}}\right)+\left(\frac{{\displaystyle 1}}{{\displaystyle 7}}-\frac{{\displaystyle 1}}{{\displaystyle 7}}\right)+\left(-\frac{{\displaystyle 1}}{{\displaystyle 8}}+\frac{{\displaystyle 1}}{{\displaystyle 8}}\right) \\ +\left(\frac{1}{9}-\frac{1}{9}\right)+\mathrm{...}+\left(-\frac{1}{98}+\frac{1}{98}\right)+\left(\frac{1}{99}-\frac{1}{99}\right)-\frac{1}{100} \\ =\frac{{\displaystyle 38}}{{\displaystyle 50}}+\frac{{\displaystyle 1}}{{\displaystyle 4}}+0+0+0+0+0+\mathrm{...}+0+0-\frac{{\displaystyle 1}}{{\displaystyle 100}} \\ =\frac{{\displaystyle 38}}{{\displaystyle 50}}+\frac{{\displaystyle 1}}{{\displaystyle 4}}-\frac{{\displaystyle 1}}{{\displaystyle 100}}=\frac{{\displaystyle 76}}{{\displaystyle 100}}+\frac{{\displaystyle 25}}{{\displaystyle 100}}-\frac{{\displaystyle 1}}{{\displaystyle 100}} \\ =\frac{{\displaystyle 76+25-1}}{{\displaystyle 100}} \\ =\frac{{\displaystyle 100}}{{\displaystyle 100}} \end{gathered}$$

= 1.

Vậy H = 1.