3A=3+3^2+3^3+....+3^20

2A=3A-A=3^20-1

A=(3^20-1)/2

A<B

Ta có : \(A=1+2+2^2+...+2^{2017}\)(1)

\(\Rightarrow2A=2+2^2+2^3+...+2^{2018}\)(2)

Lấy (2) trừ (1) ta có : 

\(\Rightarrow A=2^{2018}-1\)

\(\Rightarrow A< B\). Vì \(B=2^{2018}\)

A = 1+2+2²+2³+.....+2²⁰¹⁷

2A = 2(1+2+2²+2³+.....+2²⁰¹⁷)  = 2+2²+2³+2⁴+.....+2²⁰¹⁸

2A - A = 2+2²+2³+2⁴+.....+2²⁰¹⁸- (1+2+2²+2³+.....+2²⁰¹⁷)

=> A = 2+2²+2³+2⁴+.....+2²⁰¹⁸-1-2-2²-2³-.....-2²⁰¹⁷

       A =2²⁰¹⁸-1 < 2²⁰¹⁸

Vậy A < B

Xét: 

+) a,<b\(\Rightarrow ab+2a< ab+2b\)

\(\Leftrightarrow a\left(b+2\right)< b\left(a+2\right)\)

\(\frac{\Rightarrow a}{b}< \frac{a+2}{b+2}\)

Vd : 

a=2 , b=3 thì:

\(\frac{2}{3}< \frac{2+2}{3+2}=\frac{4}{5}\)

Tương tự xét với a> b; a=b

a: Ta có: \(A=\frac59+\left(-\frac57\right)+\left(-\frac{20}{48}\right)+\frac{8}{12}+\left(-\frac{21}{48}\right)\)

\(=\frac59-\frac57-\frac{41}{48}+\frac{32}{48}\)

\(=\frac{35-45}{63}-\frac{9}{48}=\frac{-10}{63}-\frac{3}{16}=\frac{-160-189}{63\cdot16}=\frac{-349}{1008}\)

b: \(B=\left(-\frac59\right)+\frac{8}{15}+\left(-\frac{2}{11}\right)+\left(\frac{4}{-9}\right)+\frac{2}{45}\)

\(=\left(-\frac59-\frac49\right)+\frac{8}{15}+\frac{2}{45}-\frac{2}{11}\)

\(=-1-\frac{2}{11}+\frac{24}{45}+\frac{2}{45}=-\frac{13}{11}+\frac{26}{45}=\frac{-13\cdot45+26\cdot11}{11\cdot45}=\frac{-299}{495}\)

c: \(\frac{1}{11}>\frac{1}{20};\frac{1}{12}>\frac{1}{20};\ldots;\frac{1}{20}=\frac{1}{20}\)

Do đó: \(\frac{1}{11}+\frac{1}{12}+\cdots+\frac{1}{20}>\frac{1}{20}+\frac{1}{20}+\cdots+\frac{1}{20}\)

=>S>10/20

=>S>1/2

874^2 >878*870

19^4 > 17*18*20*21

a.874^2<878.870

b.19^4<17.18.20.21

a) Ta có :

874² = 874 . 874 = 874 . (870 + 4) = 874 . 870 + 874 . 4

878 . 870 = (874 + 4) . 870 = 874 . 870 + 870 . 4

Ta thấy : 874 . 870 + 874 . 4 > 874 . 870 + 870 . 4

=> 874² > 878 . 870

câu b, tự làm

a) Ta có: 874²=874.874

               878.870 = (870+8).870 = 870.870+870.2

=> 874² > 878.870

b) Ta có: 19⁴ = 19.19.19.19 

  => 19⁴ > 17.18.19.20.21