555555555555500000000000000.................

Ta có : \(\frac{2017.2018+1}{2017.2018}=1+\frac{1}{2017.2018}\)

             \(\frac{2018.2019+1}{2018.2019}=1+\frac{1}{2018.2019}\)

Mà : \(\frac{1}{2017.2018}>\frac{1}{2018.2019}\) => \(\frac{2017.2018+1}{2017.2018}>\frac{2018.2019+1}{2018.2019}\)

Ta có:

\(C=\frac{2017.2018-1}{2017.2018}=1-\frac{1}{2017.2018}\)

\(D=\frac{2018.2019-1}{2018.2019}=1-\frac{1}{2018.2019}\)

Mà ta có:

\(\frac{1}{2017.2018}>\frac{1}{2018.2019}\Rightarrow1-\frac{1}{2017.2018}< 1-\frac{1}{2018.2019}\Rightarrow C< D\)

\(\frac{2017.2018-1}{2017.2018}=1-\frac{1}{2017.2018}\)

\(\frac{2018.2019-1}{2018.2019}=1-\frac{1}{2018.2019}\)

Ta thấy      \(2017.2018< 2018.2019\)

nên      \(\frac{1}{2017.1018}>\frac{1}{2018.2019}\)

\(\Rightarrow\)\(1-\frac{1}{2017.2018}< 1-\frac{1}{2018.2019}\)

Vậy      \(\frac{2017.2018-1}{2017.2018}< \frac{2018.2019-1}{2018.2019}\)

a) \(\frac{53}{57}=\frac{530}{570}\)

Ta có : 1 - \(\frac{530}{570}\)= \(\frac{40}{570}\) ;    1 - \(\frac{531}{571}=\frac{40}{571}\)

Vì \(\frac{40}{570}>\frac{40}{571}\) nên \(\frac{53}{57}< \frac{531}{571}\)

(1.2 + 2.3 + 3.4 + ... + 2018.2019) - (1² + 2² + ... + 2018²)

= (1.2 + 2.3 + ... + 2018.2019) - (1.1 + 2.2 + ... + 2018.2018)

= (1.2 + 2.3 + ... + 2018.2019) - [1.(2 - 1) + 2.(3 - 1) + ... + 2018.(2019 - 1)]

= (1.2 + 2.3 + ... + 2018.2019) - (1.2 + 2.3 + ... + 2018.2019 - 1 - 2 - 3 - ... - 2018)

= (1.2 + 2.3 + ... + 2018.2019) - [1.2 + 2.3 + ... + 2018.2019 - (1 + 2 + ... + 2018)]

= (1.2 + 2.3 + ... + 2018.2019) - (1.2 + 2.3 + ... + 2018.2019) + (1 + 2 + 3 + ... + 2018)

= 1 + 2 + ... + 2018 (có : (2018 - 1) : 1 + 1 = 2018 (số))

= (2018 + 1).2018 : 2

= 2037171

cảm ơn nhé

\(S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2017.2018}+\frac{1}{2018.2019}\)

\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2017}-\frac{1}{2018}+\frac{1}{2018}-\frac{1}{2019}\)

( gạch bỏ các phân số giống nhau)

\(S=1-\frac{1}{2019}\)

\(S=\frac{2018}{2019}\)

CHÚC BN HỌC TỐT!!!!

S=1/1.2+1/2.3+1/3.4+............1/2017.2018+1/2018.2019

S=1/2.(1+1/3.2+1/3.2+.............1/2017.1009+1/1009.2019)

S=1/4.(2+2/3.2+2/3.2+..............2/2017.1009+2/1009.2019)

S=1/4.(1-1/2+1/2-1/3+1/3+..........+1/1009-1/1009+1/2019)

S=1/4.(1-1/2019)

S=1/4.2018/2019=1009/4038

a) Ta có : 

N = 2018 + 2019/2019 + 2020

   = 2018/2019 + 2020   +    2019/2019 + 2020

Ta thấy : 2018/2019 + 2020  <  2018/2019 ( Vì 2019 + 2020 > 2019 )

              2019/2019 + 2020  < 2019/2020 ( Vì 2019 + 2020 > 2020 )

=>  2018/2019 + 2020   +    2019/2019 + 2020  <   2018/2019  +  2019/2020

=> M > N

b) Mk ko bt làm !!

c) Ta có :

  19/31 > 1/2

  17/35 < 1/2

=> 19/31 > 17/35

d) Ta có :

   3535/3434 = 1 + 1/3534

   2323/2322 = 1 + 1/2322

Ta thấy : 

1/3534 < 1/2322 ( Vì 3534 > 2322 )

=> 1 + 1/3534 < 1 + 1/2322

=> 3535/3534 < 2323/2322

Hok tốt !