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Ta có:
\(\frac{2015^2-2014^2}{2015^2+2014^2}-\frac{\left(2015-2014\right)^2}{\left(2015+2014\right)^2}\)
\(=\frac{2015+2014}{2015^2+2014^2}-\frac{1}{\left(2015+2014\right)^2}\)
Ta thấy phân số thứ nhất có tử lớn hơn phân số thứ 2 và có mẫu bé hơn nên phân số thứ nhất > phâm số thứ 2
Hay \(\frac{2015^2-2014^2}{2015^2+2014^2}>\frac{\left(2015-2014\right)^2}{\left(2015+2014\right)^2}\)
\(C=\dfrac{2014\left(2015^2+2016\right)-2016\left(2015^2-2014\right)}{2014\left(2013^2-2012\right)-2012\left(2013^2+2014\right)}\)
\(=\dfrac{2.2014.2016+2014.2015^2-2016.2015^2}{2014.2013^2-2012.2013^2-2.2012.2014}\)
\(=\dfrac{2.\left(2015+1\right)\left(2015-1\right)-2.2015^2}{2.2013^2-2.\left(2013+1\right)\left(2013-1\right)}\)
\(=\dfrac{2.\left(2015^2-1\right)-2.2015^2}{2.2013^2-2.\left(2013^2-1\right)}=\dfrac{-2}{2}=-1\)
Ta có:
\(M=\frac{x\left(yz-x^2\right)+y\left(zx-y^2\right)+z\left(xy-z^2\right)}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}=\frac{xyz-x^3+xyz-y^3+xyz-z^3}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}=\frac{3xyz-x^3-y^3-z^3}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}\)
\(-M=\frac{x^3+y^3+z^3-3xyz}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}\)
Xét đẳng thức phụ:
\(a^3+b^3+c^3-3abc=\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc=\left[\left(a +b\right)^3+c^3\right]-3ab\left(a+b+c\right)\)\(=\left(a+b+c\right)\left(\left(a+b\right)^2-c\left(a+b\right)+c^2\right)-ab\left(a+b+c\right)\)
\(=\left(a+b+c\right)\left[\left(a+b\right)^2-c\left(a+b\right)+c^2-ab\right]=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)\)
\(=\frac{1}{2}\left(a+b+c\right)\left(2a^2+2b^2+2c^2-2ab-abc-ac\right)\)
\(=\frac{1}{2}\left(a+b+c\right)\left[\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\right]\)
Thay vào -M ta có:
\(-M=\frac{\frac{1}{2}\left(x+y+z\right)\left[\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\right]}{\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2}=\frac{1}{2}\left(x+y+z\right)\Rightarrow M=-\frac{1}{2}\left(x+y+z\right)\)
Giờ thay: \(x=2014^{2015}-20142015;y=20142015-2015^{2014};z=2015^{2014}-2014^{2015}\)
Ta có:
\(M=-\frac{1}{2}\left(2014^{2015}-20142015+20142015-2015^{2014}+2015^{2014}-2014^{2015}\right)=0\)
\(VT=\dfrac{2}{x}-\dfrac{2}{x+1}+\dfrac{2}{x+1}-\dfrac{2}{x+2}+...+\dfrac{2}{x+2014}-\dfrac{2}{x+2015}\)
\(VT=\dfrac{2}{x}-\dfrac{2}{x+2015}=\dfrac{2\left(x+2015-x\right)}{x\left(x+2015\right)}=\dfrac{4030}{x\left(x+2015\right)}\)
b: \(=\dfrac{2014\cdot2015^2+2014\cdot2016-2016\cdot2015^2+2016\cdot2014}{2014\cdot2013^2-2014\cdot2012-2012\cdot2013^2-2012\cdot2014}\)
\(=\dfrac{2015^2\cdot\left(-2\right)+2\cdot\left(2015^2-1\right)}{2013^2\cdot\left(-2\right)-2\cdot\left(2013^2-1\right)}\)
\(=\dfrac{\left(-2\right)\cdot\left(2015^2-2015^2+1\right)}{\left(-2\right)\cdot\left(2013^2+2013^2-1\right)}=\dfrac{1}{2\cdot2013^2}\)
\(\text{Có }:\left(\dfrac{2015-2014}{2015+2014}\right)^2=\dfrac{\left(2015-2014\right)^2}{2015^2+2\cdot2015\cdot2014+2014^2}\\ \dfrac{2015^2-2014^2}{2015^2+2014^2}=\dfrac{\left(2015-2014\right)\left(2015+2014\right)}{2015^2+2014^2}\)
\(\text{Do }2015-2014< 2015+2014\\ \Rightarrow\left(2015-2014\right)^2< \left(2015+2014\right)\left(2015-2014\right)\\ \Rightarrow\dfrac{\left(2015-2014\right)^2}{2015^2+2\cdot2015\cdot2014+2014^2}< \dfrac{\left(2015+2014\right)\left(2015-2014\right)}{2015^2+2\cdot2015\cdot2014+2014^2}\)
\(\text{Mà }2015^2+2\cdot2015\cdot2014+2014^2>2015^2+2014^2\\ \Rightarrow\dfrac{\left(2015+2014\right)\left(2015-2014\right)}{2015^2+2\cdot2015\cdot2014+2014^2}< \dfrac{\left(2015+2014\right)\left(2015-2014\right)}{2015^2+2014^2}\left(2\right)\)
Từ \(\left(1\right)\) và \(\left(2\right)\Rightarrow\dfrac{\left(2015-2014\right)^2}{2015^2+2\cdot2015\cdot2014+2014^2}< \dfrac{\left(2015+2014\right)\left(2015-2014\right)}{2015^2+2014^2}\)
\(\Rightarrow\left(\dfrac{2015-2014}{\left(2015+2014\right)}\right)^2< \dfrac{2015^2-2014^2}{2015^2+2014^2}\)