Tính tổng sau: S= $\frac{1}{1.2.3}$ + $\frac{1}{2.3.4}$ + $\frac{1}{3.4.5}$ + … + $\frac{1}{98.99.100}$

Đáp án:

$S = \dfrac{4949}{19800}$

Giải thích các bước giải:

$S = \dfrac{1}{1.2.3} + \dfrac{1}{2.3.4} + \cdots + \dfrac{1}{98.99.100}$

$\Leftrightarrow 2S = \dfrac{2}{1.2.3} + \dfrac{2}{2.3.4} + \cdots + \dfrac{2}{98.99.100}$

$\Leftrightarrow 2S = \dfrac{3 – 1}{1.2.3} + \dfrac{4 – 2}{2.3.4} + \cdots + \dfrac{100 – 98}{98.99.100}$

$\Leftrightarrow 2S = \dfrac{3}{1.2.3} – \dfrac{1}{1.2.3} + \dfrac{4}{2.3.4} – \dfrac{2}{2.3.4} + \cdots + \dfrac{100}{98.99.100} – \dfrac{98}{98.99.100}$

$\Leftrightarrow 2S = \dfrac{1}{1.2} – \dfrac{1}{2.3} + \dfrac{1}{2.3} – \dfrac{1}{3.4} + \cdots + \dfrac{1}{98.99} – \dfrac{1}{99.100}$

$\Leftrightarrow 2S = \dfrac{1}{2} – \dfrac{1}{9900}$

$\Leftrightarrow S = \dfrac{1}{4} -\dfrac{1}{19800} = \dfrac{4949}{19800}$