\(A=\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+2009}\)
\(=\frac{1}{\frac{2\cdot\left(1+2\right)}{2}}+\frac{1}{\frac{3\cdot\left(3+1\right)}{2}}+\frac{1}{\frac{4\cdot\left(4+1\right)}{2}}+...+\frac{1}{\frac{2009\cdot\left(2009+1\right)}{2}}\)
\(=\frac{2}{2\cdot3}+\frac{2}{3\cdot4}+\frac{2}{4\cdot5}+...+\frac{2}{2009\cdot2010}\)
\(=2\cdot\left(\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{2009\cdot2010}\right)\)
\(=2\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2009}-\frac{1}{2010}\right)\)
\(=2\cdot\left(\frac{1}{2}-\frac{1}{2010}\right)\)
\(=1-\frac{1}{1005}\)
\(=\frac{1004}{1005}\)
1/1+2=3=1/1+2+2=6=1/1+2+3+4=10+3+6=19+1/1+2+3+4=29+3+6+10+19+2009=2076nếu mình làm sai thì nhớ chỉ dùm
nhớ kết bạn với mình nhé
\(\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+4+...+2009}\)
\(=\frac{1}{\left(2+1\right).2:2}+\frac{1}{\left(3+1\right).3:2}+\frac{1}{\left(4+1\right).4:2}+...+\frac{1}{\left(2009+1\right).2009:2}\)
\(=\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{2009.2010}=2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2009}-\frac{1}{2010}\right)\)\(=2.\left(\frac{1}{2}-\frac{1}{2010}\right)=\frac{2.502}{1005}=\frac{1004}{1005}\)
kết quả đúng \(\frac{1004}{1005}\)
nhớ k nha hok tốt
