A = (n + 2015)(n + 2016) + n² + n

= (n + 2015)(n + 2015 + 1) + n(n + 1)

Tích 2 số tự nhiên liên tiếp luôn chia hết cho 2

=> (n + 2015)(n + 2015 + 1) chia hết cho 2

      n(n + 1) chia hết cho 2

=> (n + 2015)(n + 2015 + 1) + n(n + 1) chia hết cho 2

=> A chia hết cho 2 với mọi n \(\in\) N (đpcm)

Tạm thời chỉ nghĩ ra được cách này -_- 

Ta có : 

\(A=\frac{2014}{2015}+\frac{2015}{2016}+\frac{2016}{2014}\)

\(A=\frac{2015-1}{2015}+\frac{2016-1}{2016}+\frac{2014+2}{2014}\)

\(A=\frac{2015}{2015}-\frac{1}{2015}+\frac{2016}{2016}-\frac{1}{2016}+\frac{2014}{2014}+\frac{2}{2014}\)

\(A=1-\frac{1}{2015}+1-\frac{1}{2016}+1+\frac{2}{2014}\)

\(A=\left(1+1+1\right)-\left(\frac{1}{2015}+\frac{1}{2016}-\frac{2}{2014}\right)\)

\(A=3-\left[\left(\frac{1}{2015}+\frac{1}{2016}\right)-\left(\frac{1}{2014}+\frac{1}{2014}\right)\right]\)

Lại có : 

\(\frac{1}{2015}< \frac{1}{2014}\)

\(\frac{1}{2016}< \frac{1}{2014}\)

\(\Rightarrow\)\(\frac{1}{2015}+\frac{1}{2016}< \frac{1}{2014}+\frac{1}{2014}\)

\(\Rightarrow\)\(\left(\frac{1}{2015}+\frac{1}{2016}\right)-\left(\frac{1}{2014}+\frac{1}{2014}\right)< 0\)

\(\Rightarrow\)\(A=3-\left[\left(\frac{1}{2015}+\frac{1}{2016}\right)-\left(\frac{1}{2014}+\frac{1}{2014}\right)\right]>3\)

Vậy \(A>3\)

Chúc bạn học tốt ~ 

\(A=\dfrac{2014}{2015}+\dfrac{2015}{2016}+\dfrac{2016}{2017}+\dfrac{2017}{2014}\\ =1-\dfrac{1}{2015}+1-\dfrac{1}{2016}+1-\dfrac{1}{2017}+1+\dfrac{1}{2014}+\dfrac{1}{2014}+\dfrac{1}{2014}\\ =\left(1+1+1+1\right)+\left[-\left(\dfrac{1}{2015}-\dfrac{1}{2014}+\dfrac{1}{2016}-\dfrac{1}{2014}+\dfrac{1}{2017}-\dfrac{1}{2014}\right)\right]\\ =4+\left[-\left(\dfrac{1}{2015}-\dfrac{1}{2014}+\dfrac{1}{2016}-\dfrac{1}{2014}+\dfrac{1}{2017}-\dfrac{1}{2014}\right)\right]\)

Vì \(\dfrac{1}{2015}< \dfrac{1}{2014}\), \(\dfrac{1}{2016}< \dfrac{1}{2014}\), \(\dfrac{1}{2017}< \dfrac{1}{2014}\)

\(\Rightarrow\left(\dfrac{1}{2015}-\dfrac{1}{2014}+\dfrac{1}{2016}-\dfrac{1}{2014}+\dfrac{1}{2017}-\dfrac{1}{2014}\right)< 0\\ \Rightarrow-\left(\dfrac{1}{2015}-\dfrac{1}{2014}+\dfrac{1}{2016}-\dfrac{1}{2014}+\dfrac{1}{2017}-\dfrac{1}{2014}\right)\\>0\\ \Rightarrow4+\left[-\left(\dfrac{1}{2015}-\dfrac{1}{2014}+\dfrac{1}{2016}-\dfrac{1}{2014}+\dfrac{1}{2017}-\dfrac{1}{2014}\right)\right]>4\)

Ta có: 2014/2015>2014/(2015+2016)

2015/2016>2015/(2015+2016)

=>2014/2015+2015/2016>2014/(2015+2016)+2015/(2015+2016)

hay 2014/2015+2015/2016>(2014+2015)/(2015+2016)

hay A>B

Vậy A>B

Đề là j vậy bạn?

de kieu j

sem lai di

Dấu < nhé!

2014+2015+2016/2015+2016+2017<2014/2015+2015/2016+2016/2017

dấu = đấy

de ot la dau = nha

M~1+1+1=3

N~1

=> M>N

m=n m>n m<n 1 trong 3 chắc chắn đúng ahihi =)))
 

\(y=\frac{2014}{\frac{2015}{\frac{2015}{2016}}}=\frac{2014}{2015}.\frac{2015}{2016}=\frac{1007}{1008}=1-\frac{1}{2008}\)

\(\frac{2014}{2015}=1-\frac{1}{2015}\)

Vì \(\frac{1}{2008}>\frac{1}{2015}\)nên \(\frac{1007}{1008}< \frac{2014}{2015}\)

Vậy A>y

y < 1 < A. 

Bạn chứng minh điều đó nhé!