Bài 1

\(\frac{2017}{2018}+\frac{2018}{2019}\)và \(\left(\frac{2017+2018}{2018+2019}\right)\)mk chữa lại đề luôn đó 

Ta tách :

\(\frac{2017}{\left(2018+2019\right)+2018}\)

đến đây ta tách 

\(\frac{2017}{2018+2019}< \frac{2017}{2018}\)

vậy....

mấy câu khác tương tự 

2) \(\frac{\frac{1}{2003}+\frac{1}{2004}+\frac{1}{2005}}{\frac{2}{2003}+\frac{2}{2004}+\frac{2}{2005}}\)

= \(\frac{\frac{1}{2003}+\frac{1}{2004}+\frac{1}{2005}}{2.\frac{1}{2003}+2.\frac{1}{2004}+2.\frac{1}{2005}}\)

=\(\frac{1\left(\frac{1}{2003}+\frac{1}{2004}+\frac{1}{2005}\right)}{2.\left(\frac{1}{2003}+\frac{1}{2004}+\frac{1}{2005}\right)}\)

= \(\frac{1}{2}\)

3) \(2013+\left(\frac{2013}{1+2}\right)+\left(\frac{2013}{1+2+3}\right)+...+\left(\frac{2013}{1+2+3+...+2012}\right)\)

= \(2013.\left(1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+...+2012}\right)\)

= \(2013.\left(1+\frac{1}{3}+\frac{1}{6}+...+\frac{1}{2025078}\right)\)

= \(2013.2.\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{4050156}\right)\)

=\(4026.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2012.2013}\right)\)

= \(4026.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2012}-\frac{1}{2013}\right)\)

= \(4026.\left(1-\frac{1}{2013}\right)\)

= \(4026.\frac{2012}{2013}\)

=\(4024\)

\(C=5^{2018}+\frac{1}{5^{2017}+1}=\left(5^{2017}+1\right)+\frac{1}{5^{2017}+1}\)

\(D=5^{2018}+\frac{1}{5^{2018}+1}=\left(5^{2017}+1\right)+\left(1+\frac{1}{5^{2017}+2}\right)\)

Do \(\frac{1}{5^{2017}+1}< 1+\frac{1}{5^{2017}+2}\)

Nên \(C< D\)

Ta có : C = \(\frac{5^{2018}+1}{5^{2017}+1}\)

=> \(\frac{C}{5}=\frac{5^{2018}+1}{5^{2018}+5}=1-\frac{4}{5^{2018}+5}\)

Lại có D = \(\frac{5^{2019}+1}{5^{2018}+1}\)

=> \(\frac{D}{5}=\frac{5^{2019}+1}{5^{2019}+5}=1-\frac{4}{5^{2019}+5}\)

Vì \(\frac{4}{5^{2018}+5}>\frac{4}{5^{2019}+5}\Rightarrow1-\frac{4}{5^{2018}+5}< 1-\frac{4}{5^{2019}+5}\Rightarrow\frac{C}{5}< \frac{D}{5}\Rightarrow C< D\)

Ta có : \(C=\frac{5^{2018}+1}{5^{2017}+1}\)

\(\Rightarrow\frac{1}{5}C=\frac{5^{2018}+1}{5^{2018}+5}=\frac{5^{2018}+5-4}{5^{2018}+5}=1-\frac{4}{5^{2018}+5}\)

Ta lại có : \(D=\frac{5^{2019}+1}{5^{2018}+1}\)

\(\Rightarrow\frac{1}{5}D=\frac{5^{2019}+1}{5^{2019}+5}=\frac{5^{2019}+5-4}{5^{2019}+5}=1-\frac{4}{5^{2019}+5}\)

Vì \(\frac{4}{5^{2018}+5}>\frac{4}{5^{2019}+5}\) nên \(1-\frac{4}{5^{2018}+5}< 1-\frac{4}{5^{2019}+5}\)\(\Rightarrow\frac{1}{5}C< \frac{1}{5}D\)

\(\Rightarrow C< D\)

Vậy \(C< D\).

A>B do A>4 cònB<4

Ta có : 

\(\frac{2016}{2017}>\frac{2016}{2017+2018+2019}\)

\(\frac{2017}{2018}>\frac{2017}{2017+2018+2019}\)

\(\frac{2018}{2019}>\frac{2018}{2017+2018+2019}\)

\(\Rightarrow\frac{2016}{2017}+\frac{2017}{2018}+\frac{2018}{2019}>\) \(\frac{2016}{2017+2018+2019}+\frac{2017}{2017+2018+2019}+\frac{2018}{2017+2018+2019}\)

\(\Rightarrow P>\frac{2016+2017+2018}{2017+2018+2019}\)

\(\Rightarrow P>Q\)

Chúc bạn học tốt !!! 

vì P có các số bé hơn 1 còn Q có các số lớn hơn 1 =>P<Q

Vậy P<Q.

mình làm hơi tắt xin bạn thông cảm bạn tự viết các số có trong P;Q ra nhá

Đơn giản P < Q

Vì Nhìn sơ qua ta thấy tổng P gồm các phân số bé hơn 1

Tổng Q có 3 phân số lớn hơn 1

Ta có : \(0< \frac{2017}{2018}< 1\) nên   \(\frac{2017}{2018}>\frac{2017+2019}{2018+2019}\)(1)

\(0< \frac{2018}{2019}< 1\) nên \(\frac{2018}{2019}>\frac{2018+2018}{2018+2019}\) (2)

Cộng vế theo vế 1 và 2 ta được : \(B=\frac{2017}{2018}+\frac{2018}{2019}>\frac{2017+2018+2018+2019}{2018+2019}=\frac{2017+2018}{2018 +2019}+1=A+1>A\)

Vậy B>A