Question

tính tổng: 

\(S=1+\frac{1}{1+2}+\frac{1}{1+2+3}+............+\frac{1}{1+2+3+.....+2011}\)

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Answer 1

\(S=1+\frac{1}{1+2}+\frac{1}{1+2+3}+..+\frac{1}{1+2+3+..+2011}\)

\(S=1+\frac{1}{2.\left(2+1\right):2}+\frac{1}{3.\left(3+1\right):2}+...+\frac{1}{2011.\left(2011+1\right):2}\)

\(S=1+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{2011.2012}\)

\(S=1+2\left(\frac{1}{2}-\frac{1}{\cdot3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{2011}-\frac{1}{2012}\right)\)

\(S=1+2\left(\frac{1}{2}-\frac{1}{2012}\right)\)

\(S=1+2.\frac{1}{2}-2.\frac{1}{2012}\)

\(S=1+1-\frac{1}{1006}\)

\(S=\frac{2011}{1006}\)

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