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\(\frac{x+1}{2017}+\frac{x+2}{2016}=\frac{x+3}{2015}+\frac{x+4}{2014}\)
\(\Leftrightarrow\frac{x+1}{2017}+1+\frac{x+2}{2016}+1=\frac{x+3}{2015}+1+\frac{x+4}{2014}+1\)
\(\Leftrightarrow\frac{x+2018}{2017}+\frac{x+2018}{2016}-\frac{x+2018}{2015}-\frac{x+2018}{2014}=0\)
\(\Leftrightarrow\left(x+2018\right)\left(\frac{1}{2017}+\frac{1}{2016}-\frac{1}{2015}-\frac{1}{2014}\ne0\right)=0\Leftrightarrow x=-2018\)
\(4^{2017}:\left(4^{2014}+3\cdot4^{2014}\right)\)
\(=4^{2017}:4^{2014}\left(1+3\right)\)
\(=4^3\cdot4\)
\(=4^4\)
\(=256\)
\(A=\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}+\frac{2018}{2015}\)
\(=\left(1-\frac{1}{2016}\right)+\left(1-\frac{1}{2017}\right)+\left(1-\frac{1}{2018}\right)+\left(1+\frac{3}{2015}\right)\)
\(=1-\frac{1}{2016}+1-\frac{1}{2017}+1-\frac{1}{2018}+1+\frac{1}{2015}+\frac{1}{2015}+\frac{1}{2015}\)
\(=\left(1+1+1+1\right)+\left(\left(\frac{1}{2015}-\frac{1}{2016}\right)+\left(\frac{1}{2015}-\frac{1}{2017}\right)+\left(\frac{1}{2015}-\frac{1}{2018}\right)\right)\)
\(=4+\left(\frac{1}{2015}-\frac{1}{2016}\right)+\left(\frac{1}{2015}-\frac{1}{2017}\right)+\left(\frac{1}{2015}-\frac{1}{2018}\right)\)
Vì \(\frac{1}{2015}>\frac{1}{2016};\frac{1}{2015}>\frac{1}{2017};\frac{1}{2015}>\frac{1}{2018}\)
\(\Rightarrow\frac{1}{2015}-\frac{1}{2016}>0;\frac{1}{2015}-\frac{1}{2017}>0;\frac{1}{2015}-\frac{1}{2018}>0\)
\(\Rightarrow\left(\frac{1}{2015}-\frac{1}{2016}\right)+\left(\frac{1}{2015}-\frac{1}{2017}\right)+\left(\frac{1}{2015}-\frac{1}{2018}\right)>0\)
\(\Rightarrow4+\left(\frac{1}{2015}-\frac{1}{2016}\right)+\left(\frac{1}{2015}-\frac{1}{2017}\right)+\left(\frac{1}{2015}-\frac{1}{2018}\right)>4\)
\(\Rightarrow A>4\left(dpcm\right)\)
Bài 3
\(\frac{n+6}{n+1}=\frac{n+1+5}{n+1}=\frac{n+1}{n+1}+\frac{5}{n+1}\)
\(=1+\frac{5}{n+1}\)
Vậy để \(\frac{n+6}{n+1}\in Z\Rightarrow1+\frac{5}{n+1}\in Z\)
Hay \(\frac{5}{n+1}\in Z\)\(\Rightarrow n+1\inƯ_5\)
\(Ư_5=\left\{1;-1;5;-5\right\}\)
* \(n+1=1\Rightarrow n=0\)
* \(n+1=-1\Rightarrow n=-2\)
* \(n+1=5\Rightarrow n=4\)
* \(n+1=-5\Rightarrow n=-6\)
Vậy \(n\in\left\{0;-2;4;-6\right\}\)
Bài 2:
\(\frac{10}{3.8}+\frac{10}{8.13}+\frac{10}{13.18}+\frac{10}{18.23}+\frac{10}{23.28}=2\left(\frac{1}{3}-\frac{1}{8}+\frac{1}{8}-\frac{1}{13}+...+\frac{1}{23}-\frac{1}{28}\right)\\ =2\left(\frac{1}{3}-\frac{1}{28}\right)\\ =2.\frac{56}{84}\\ =\frac{56}{42}=\frac{28}{21}\)
S1 = 1-2+3-4+....+2017-2018
= (-1)+(-1)+....+(-1)
= (-1) x 1009
= -1009
S3=2019 nha, mình ko kip viết cách giai