trước tiên bạn phải tính:

2013/1+2012/2+2011/3+.....+2/2012+1/2013

=1+2012/2)+(1+2011/3)+.....+(1+2/2012)+(1+1/2013) +1 {BƯỚC NÀY TÁCH 2013 RA LÀM 2013SỐ1 ĐỂ CÔNG VS CÁC THỪA SỐ CÒN LẠI}

=2014/2+2014/3+...+2014/2012+2014/2013+2014/2014

=2014.(1/2+1/3+....+1/2012+1/20131/2014

suy ra x=2014

bố tớ làm giáo viên, bảo bài này đúng đó

Ta có: \(M=1\cdot2014+2\cdot2013+\cdots+2014\cdot1\)

\(=2\left(1\times2014+2\times2013+\cdots+1007\times1008\right)\)

\(=2\left\lbrack1\times\left(2015-1\right)+2\times\left(2015-2\right)+\cdots+1007\times\left(2015-1007\right)\right\rbrack\)

\(=2\cdot\left\lbrack2015\times\left(1+2+\cdots+1007\right)-\left(1^2+2^2+\cdots+1007^2\right)\right\rbrack\)

\(=2\cdot\left\lbrack2015\times1007\times\frac{1008}{2}-\frac{1007\times\left(1007+1\right)\times\left(2\times1007+1\right)}{6}\right\rbrack\)

\(=2\cdot\left\lbrack2015\times1007\times504-1007\times168\times2015\right\rbrack=2\times2015\times1007\times168\left(3-1\right)=4\times168\times2015\times1007\)

\(N=1+\left(1+2\right)+\cdots+\left(1+2+\cdots+2014\right)\)

\(\) \(=\frac{1\times2}{2}+\frac{2\times3}{2}+\cdots+\frac{2014\times2015}{2}\)

\(=\frac12\times\left(1\times2+2\times3+\cdots+2014\times2015\right)\)

\(=\frac12\times\left\lbrack1\times\left(1+1\right)+2\times\left(2+1\right)+\cdots+2014\times\left(2014+1\right)\right\rbrack\)

\(=\frac12\times\left\lbrack\left(1\times1+2\times2+\cdots+2014\times2014\right)+\left(1+2+\cdots+2014\right)\right\rbrack\)

\(=\frac12\times\left\lbrack\frac{2014\times\left(2014+1\right)\times\left(2\times2014+1\right)}{6}+\frac{2014\times2015}{2}\right\rbrack\)

\(=\frac12\times\left\lbrack1007\times2015\times1343+1007\times2015\right\rbrack=\frac12\times1007\times2015\times\left(1343+1\right)\)

=1007x2015x672

=4x168x2015x1007

Do đó: M=N

\(\frac{\frac{1}{2012}+\frac{1}{2013}-\frac{1}{2014}}{\frac{5}{2012}+\frac{5}{2013}-\frac{5}{2014}}-\frac{\frac{2}{2013}+\frac{2}{2014}-\frac{2}{2015}}{\frac{3}{2013}+\frac{3}{2014}-\frac{3}{2015}}\)

=\(\frac{\frac{1}{2012}+\frac{1}{2013}-\frac{1}{2014}}{5\left(\frac{1}{2012}+\frac{1}{2013}-\frac{1}{2014}\right)}-\frac{2\left(\frac{1}{2013}+\frac{1}{2014}-\frac{1}{2015}\right)}{3\left(\frac{1}{2013}+\frac{1}{2014}-\frac{1}{2015}\right)}=\frac{1}{5}-\frac{2}{3}=\frac{3}{15}-\frac{10}{15}=-\frac{7}{15}\)

Bạn hỏi hay trả lời luôn dzậy?

\(\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}\right)x=\dfrac{2013}{1}+\dfrac{2012}{2}+...+\dfrac{2}{2012}+\dfrac{1}{2013}\)

\(\Leftrightarrow\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}\right)x=\left(1+\dfrac{2012}{2}\right)+\left(1+\dfrac{2011}{3}\right)+...+\left(1+\dfrac{2}{2012}\right)+\left(1+\dfrac{1}{2013}\right)+1\)

\(\Leftrightarrow\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}\right)x=\dfrac{2014}{2}+\dfrac{2014}{3}+...+\dfrac{2014}{2012}+\dfrac{2014}{2013}+\dfrac{2014}{2014}\)

\(\Leftrightarrow\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}\right)x=2014.\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2012}+\dfrac{1}{2013}+\dfrac{1}{2014}\right)\)

\(\Leftrightarrow x=\dfrac{2014.\left(\dfrac{1}{2}+\dfrac{1}{3}+....+\dfrac{1}{2014}\right)}{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}}\)

\(\Leftrightarrow x=2014\)

Vậy \(x=2014\)

\(VP=\dfrac{2013}{1}+\dfrac{2012}{2}+...+\dfrac{1}{2013}\\ =\dfrac{2012}{2}+1+\dfrac{2011}{3}+1+...+\dfrac{1}{2013}+1+1\\ =\dfrac{2014}{2}+\dfrac{2014}{3}+...+\dfrac{2014}{2013}+\dfrac{2014}{2014}\\ =2014\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}\right)\)

\(\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}\right)x=2014\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}\right)\\ x=2014\)

Vậy x = 2014

biết vậy mà vẫn đòi lấy ảnh! ok!