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\(\frac{x_1-1}{2010}=\frac{x_2-2}{2009}=...=\frac{x_{2010}-2010}{1}=\frac{\left(x_1-1\right)+\left(x_2-2\right)+...+\left(x_{2010}-2010\right)}{1+2+...+2010}\) (TC DTSBN)
\(=\frac{\left(x_1+x_2+...+x_{2010}\right)-\left(1+2+...+2010\right)}{1+2+...+2010}=\frac{2.\left(1+2+...+2010\right)-\left(1+2+...+2010\right)}{1+2+...+2010}=1\)
\(\Rightarrow x_1-1=2010;x_2-1=2009;....;x_{2010}-2010=1\)
=> x1 = x2 = x3 =..... = x2010 = 2011
![](https://rs.olm.vn/images/avt/0.png?1311)
Áp dụng dãy tỉ số bằng nhau ta có:
\(\frac{x_1}{x_2}=\frac{x_2}{x_3}=...=\frac{x_{2016}}{x_{2016} }=\frac{x_1+x_2+...+x_{2017}}{x_2+x_3+...+x_{2017}} \)( 2016 số)
\(=>\frac{x_1^{2016}}{x_2^{2016}}=\frac{x_2^{2016}}{ x_3^{2016}}=...=\frac{x_{2016}^{2016}}{x_{2017}^{2016}} =\frac{(x_1+x_2+...+x_{2016})^{2016}}{ (x_2+x_3+...+x_{2017})^{2016}}\)
Mà \(\frac{x_1^{2016}}{x_2^{2016}}=\frac{x_1}{x_2}. \frac{x_2}{x_3}.\frac{x_3}{x_4}...\frac{x_{2016}}{x_{2017}} =\frac{x_1}{x_{2017}}\)
=>đpcm
![](https://rs.olm.vn/images/avt/0.png?1311)
Áp dụng tính chất dãy tỉ số bằng nhau ta có :
\(\frac{x_1-1}{3}=\frac{x_2-2}{2}=\frac{x_3-3}{1}=\frac{\left(x_1-1\right)+\left(x_2-2\right)+\left(x_3-3\right)}{3+2+1}=\frac{\left(x_1+x_2+x_3\right)-6}{6}=\frac{30-6}{6}=\frac{24}{6}=4\)
=> \(\frac{x_1-1}{3}=4\Rightarrow x_1=13\)
\(\frac{x_2-2}{2}=4\Rightarrow x_2=10\)
\(\frac{x_3-3}{1}=4\Rightarrow x_3=7\)
=> \(x_1.x_2-x_2.x_3=13.10-10.7=10\left(13-7\right)=10.6=60\)
Vậy \(x_1.x_2-x_2.x_3=60\)
\(\frac{x_1-1}{2010}=...=\frac{x_{2010}-2010}{1}=\frac{x_1+x_2+...+x_{2010}-\left(1+2+...+2010\right)}{2010+2009+...+1}\)
\(=\frac{2\left(1+2+...+2010\right)-\left(1+2+...+2010\right)}{1+2+...+2010}=1\)
Vậy thay vào ta được: \(x_1=x_2=...=x_{2010}=2011\)