1/2 + 1/6 + 1/12 + 1/20+ 1/30 + ... + 1/9900
= 1/1.2 + 1/2.3 + 1/3.4 + + 1/4.5 + 1/5.6 + ... + 1/99.100
= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 + ... + 1/99 - 1/100
= 1 - 1/100
= 99/100
Đặt \(S=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+...+\frac{1}{9900}\)
\(\Rightarrow S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{99.100}\)
\(\Rightarrow S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}\)
\(\Rightarrow S=1-\frac{1}{100}\)
\(\Rightarrow S=\frac{100}{100}-\frac{1}{100}\)
\(\Rightarrow S=\frac{99}{100}\)
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