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Ta có: \(M=\frac{2017^{2015}+1}{2017^{2015}-1}=\frac{2017^{2015}-1+2}{2017^{2015}-1}=1+\frac{2}{2017^{2015}-1}\)
\(N=\frac{2017^{2015}-5}{2017^{2015}-3}=\frac{2017^{2015}-3-2}{2017^{2015}-3}=1-\frac{2}{2017^{2015}-3}\)
Vì \(\frac{2}{2017^{2015}-1}>-\frac{2}{2017^{2015}-3}\)nên M>N
Ta có :
\(2017A=\dfrac{2017\left(2017^{2015}+1\right)}{2017^{2016}+1}\)
\(=\dfrac{2017^{2016}+2017}{2017^{2016}+1}\)
\(=\dfrac{\left(2017^{2016}+1\right)+2016}{2017^{2016}+1}\)
\(=\dfrac{2017^{2016}+1}{2017^{2016}+1}\) + \(\dfrac{2016}{2017^{2016}+1}\)
\(=1+\dfrac{2016}{2017^{2016}+1}\) (1)
Tương tự :
\(2017B=\dfrac{2017\left(2017^{2014}+1\right)}{2017^{2015}+1}\)
\(=\dfrac{2017^{2015}+2017}{2017^{2015}+1}\)
\(=1+\dfrac{2016}{2017^{2016}+1}\) (2)
Từ (1) và (2) => \(2017A< 2017B\)
=> \(A< B\)
Áp dụng tính chất \(\frac{a}{b}< 1\Rightarrow\frac{a}{b}< \frac{a+m}{b+m}\)ta có:
\(B=\frac{2015^{2017}+1}{2015^{2018}+1}< \frac{2015^{2017}+1+2014}{2015^{2018}+1+2014}=\frac{2015^{2017}+2015}{2015^{2018}+2015}\)
\(=\frac{2015\left(2015^{2016}+1\right)}{2015\left(2015^{2017}+1\right)}=\frac{2015^{2016}+1}{2015^{2017}+1}\)
\(\Rightarrow\frac{2015^{2017}+1}{2015^{2018}+1}< \frac{2015^{2016}+1}{2015^{2017}+1}\)
Vậy \(B< A\)
Hay \(A>B\)
TA có :\(\frac{2015.2016-1}{2015.2016}=\frac{2015.2016}{2015.2016}-\frac{1}{2015.2016}=1-\frac{1}{2015.2016}\)
Ta có:\(\frac{2016.2017-1}{2016.2017}=\frac{2016.2017}{2016.2017}-\frac{1}{2016.2017}=1-\frac{1}{2016.2017}\)
Vì \(2015.2016< 2016.2017\)
\(\Rightarrow\frac{1}{2015.2016}>\frac{1}{2016.2017}\)
\(\Rightarrow1-\frac{1}{2015.2016}< 1-\frac{1}{2016.2017}\)
\(\Rightarrow\frac{2015.2016-1}{2015.2016}< \frac{2016.2017-1}{2016.2017}\)
Vậy \(\frac{2015.2016-1}{2015.2016}< \frac{2016.2017-1}{2016.2017}\)
Ta có:\(Q=\frac{2015+2016+2017}{2016+2017+2018}=\frac{2015}{2016+2017+2018}+\frac{2016}{2016+2017+2018}+\frac{2017}{2016+2017+2018}\)
Vì \(\hept{\begin{cases}\frac{2015}{2016}>\frac{2015}{2016+2017+2018}\\\frac{2016}{2017}>\frac{2016}{2016+2017+2018}\\\frac{2017}{2018}>\frac{2017}{2016+2017+2018}\end{cases}}\)
\(\Rightarrow\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}>\frac{2015}{2016+2017+2018}+\frac{2016}{2016+2017+2018}+\frac{2017}{2016+2017+2018}\)
\(\Rightarrow P>Q\)
Vậy P > Q
Ta có:
M=\(\dfrac{2017^{2015}+1}{2017^{2015}-1}=\dfrac{2017^{2015}-1+2}{2017^{2015}-1}=1+\dfrac{2}{2017^{2015}-1}>1\left(1\right)\)
N=\(\dfrac{2017^{2015}-5}{2017^{2015}-3}=\dfrac{2017^{2015}-3-2}{2017^{2015}-3}=1-\dfrac{2}{2017^{2015}-3}< 1\left(2\right)\)
Từ (1) và (2) suy ra M>1>N
Vậy M>N.
Ta có :
\(\dfrac{2017^{2015}+1}{2017^{2015}-1}>\dfrac{2017^{2015}}{2017^{2015}}>\dfrac{2017^{2015}-5}{2017^{2015}-3}\)
Tick mình nha bạn hiền.