Question

[trần nguyễn việt anh]

rút gọn:A=(1-1/2^2)+(1-1/3^2)+.......+(1-1/2010^2)

Answer 1

[╰Nguyễn Trí Nghĩa (team toán học)ッ...]

Hình như đề sai vì không ra được kết quả đề đúng

\(A=\left(1-\frac{1}{2^2}\right).\left(1-\frac{1}{3^2}\right).......\left(1-\frac{1}{2010^2}\right)\)

\(\Rightarrow A=\left(\frac{2^2}{2^2}-\frac{1}{2^2}\right).\left(\frac{3^2}{3^2}-\frac{1}{3^2}\right)............\left(\frac{2010^2}{2010^2}-\frac{1}{2010^2}\right)\)

\(\Rightarrow A=\frac{2^2-1}{2^2}.\frac{3^2-1}{3^2}.............\frac{2010^2-1}{2010^2}\)

\(\Rightarrow A=\frac{4-1}{2^2}.\frac{9-1}{3^2}................\frac{4040100-1}{2010^2}\)

\(\Rightarrow A=\frac{3}{2^2}.\frac{8}{3^2}..............\frac{4040099}{2010^2}\)

\(\Rightarrow A=\frac{1.3}{2.2}.\frac{2.4}{3.3}................\frac{2009.2011}{2010.2010}\)

\(\Rightarrow A=\frac{\left(1.2...............2009\right).\left(3.4..............2011\right)}{\left(2.3............2010\right).\left(2.3.............2010\right)}=\frac{1.2011}{2010.2}=\frac{2011}{4020}\)

Vậy A=\(\frac{2011}{4020}\)

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