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( 2013 x 2014 +2014 x 2015 + 2015 x 2016 ) x ( 1 + 1/3 - 1 - 1/3 )
= ( 2013 x 2014 + 2014 x 2015 + 2015 x 2016 ) x 0
= 0
\(\frac{2012+2013.2014}{2014.2015-2016}=\frac{2012+2013.2014}{2014.\left(2013+2\right)-2016}\)
\(=\frac{2012+22013.2014}{2014.2013+2014.2-2016}\)
\(=\frac{2012+2013.2014}{2014.2013+2028-2016}\)
\(=\frac{2012+2013.2014}{2014.2013+2012}=1\)
\(\frac{2012+2013.2014}{2014.2015-2016}\)
\(=\frac{2012+2013.2014}{2014.\left(2013+2\right)-2016}\)
\(=\frac{2012+2013.2014}{2014.2013+2014.2-2016}\)
\(=\frac{2012}{2014.2-2016}\)
\(=\frac{2012}{4028-2016}\)
\(=\frac{2012}{2012}\)
\(=1\)
A = \(\frac{2013}{2014}+\frac{2014}{2015}>\frac{1}{2}+\frac{1}{2}=1\)
\(B=\frac{2013+2014+2015}{2014+2015+2016}<1\)
\(Vậy:A>B\)
Đúng nha Nguyễn Bình Minh
so sánh:
\(A=\frac{2013}{2014}+\frac{2014}{2015}+\frac{2015}{2016}\) và\(B=\) \(\frac{2013+2014+2015}{2014+2015+2016}\)
\(B=\frac{2013}{2014+2015+2016}+\frac{2014}{2014+2015+2016}+\frac{2015}{2014+2015+2016}\)
Ta có: \(\frac{2013}{2014}>\frac{2013}{2014+2015+2016}\)
\(\frac{2014}{2015}>\frac{2014}{2014+2015+2016}\)
\(\frac{2015}{2016}>\frac{2015}{2014+2015+2016}\)
\(\Rightarrow\frac{2013}{2014}+\frac{2014}{2015}+\frac{2015}{2016}>\frac{2013+2014+2015}{2014+2015+2016}\)
Vậy: \(A>B\)
\(A=\left[1+\left(-2\right)\right]+\left[3+\left(-4\right)\right]+....+\left[2013+\left(-2014\right)+2015\right]\)
\(A=\left(-1\right)+\left(-1\right)+....+\left(-1\right)+2015\left(\text{1007 số hạng }\left(-1\right)\right)=1008\)
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)
\(\frac{2014}{2015}\) +\(\frac{2015}{2016}\) < 2014+\(\frac{2015}{2015}\) +2016
\(\frac{2014\cdot2015+2014}{2014\cdot2016-1000}\)
\(=\frac{2014\cdot2015+2014}{2014\cdot\left(2015+1\right)-1000}\)
\(=\frac{2014\cdot2015+2014}{2014\cdot2015+2014\cdot1-1000}\)
\(=\frac{2014\cdot2015+2014}{2014\cdot2015+1014}\)
\(=\frac{2014}{1014}=.......\)
2014.2015 + 2014 = 4028.2015
2014.2016 - 1000 = 1014.2016