Ta có : 

\(A=\frac{1}{2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{18.19.20}\)

\(\Rightarrow A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{18.19.20}\)

\(\Rightarrow A=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)

\(\Rightarrow A=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{19.20}\right)\)

\(\Rightarrow A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{380}\right)\)

\(\Rightarrow A=\frac{1}{4}-\frac{1}{760}< \frac{1}{4}\)

Vậy \(A< \frac{1}{4}\)

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)

\(\Rightarrow A=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)

\(\Rightarrow A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{380}\right)=\frac{1}{2}\left(\frac{189}{380}\right)=\frac{189}{760}< \frac{1}{4}\)

Ta có: \(A=\frac{1}{2\times3}+\frac{1}{2\times3\times4}+\frac{1}{3\times4\times5}+....+\frac{1}{18\times19\times20}\)

              \(=\frac{1}{2}\times\left(\frac{1}{1\times2}-\frac{1}{2\times3}+\frac{1}{2\times3}-\frac{1}{3\times4}+...+\frac{1}{18\times19}-\frac{1}{19\times20}\right)\)

               \(=\frac{1}{2}\times\left(\frac{1}{1\times2}-\frac{1}{19\times20}\right)\)

                 \(=\frac{1}{2}\times\frac{1}{1\times2}-\frac{1}{2}\times\frac{1}{19\times20}\)

                   \(=\frac{1}{4}-\frac{1}{2}\times\frac{1}{19\times20}< \frac{1}{4}\)

Vậy A < 1/4

Lời giải:

$A=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2021}}$

$2A=1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^{2020}}$

$\Rightarrow 2A-A=1-\frac{1}{2^{2021}}$

$\Rightarrow A=1-\frac{1}{2^{2021}}

$B=\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{60}=\frac{4}{5}=1-\frac{1}{5}$

Hiển nhiên $\frac{1}{2^{2021}}< \frac{1}{5}\Rightarrow 1-\frac{1}{2^{2021}}> 1-\frac{1}{5}$

$\Rightarrow A> B$

a)

\(\dfrac{48}{92}=\dfrac{48:4}{92:4}=\dfrac{12}{23}\)
\(\dfrac{36}{69}=\dfrac{36:3}{69:3}=\dfrac{12}{23}\)
Ta có: 
Mẫu số chung 2 phân số: 23
Vì \(12=12\) nên \(\dfrac{12}{23}=\dfrac{12}{23}\)
Vậy \(\dfrac{48}{92}=\dfrac{36}{69}\)

b)

\(\dfrac{3}{5}+\dfrac{4}{7}+\dfrac{7}{8}=\dfrac{573}{280}\)

Ta có: 
Mẫu số chung 2 phân số: 280
\(\dfrac{3}{1}=\dfrac{3*280}{1*280}=\dfrac{840}{280}\)
Vì \(573< 840\) nên \(\dfrac{573}{280}< \dfrac{840}{280}\)
Vậy \(\dfrac{573}{280}< \dfrac{3}{1}\)

c)
Ta có: 
Mẫu số chung 2 phân số: 10
\(\dfrac{2}{5}=\dfrac{2*2}{5*2}=\dfrac{4}{10}\)
Vì \(1< 4\) nên\(\dfrac{1}{10}< \dfrac{4}{10}\)
Vậy \(\dfrac{1}{10}< \dfrac{2}{5}\)

d)
\(\dfrac{4}{10}=\dfrac{4:2}{10:2}=\dfrac{2}{5}\)
Ta có: 
Mẫu số chung 2 phân số: 5
Vì \(2=2\) nên \(\dfrac{2}{5}=\dfrac{2}{5}\)
Vậy \(\dfrac{4}{10}=\dfrac{2}{5}\)

Hãy quy đồng hộ mình và đánh dấu nha 

a) \(\dfrac{3}{4}=0.75\)

b) \(-2\dfrac{1}{2}=\dfrac{-5}{2}< \dfrac{-5}{3}\)

Bài 1:

a) \(\dfrac{-17}{36}\) và \(\dfrac{23}{-48}\) 

\(\dfrac{-17}{36}=\dfrac{-17.4}{36.4}=\dfrac{-68}{144}\) 

\(\dfrac{23}{-48}=\dfrac{-23}{48}=\dfrac{-23.3}{144.3}=\dfrac{-69}{144}\) 

Vì \(\dfrac{-68}{144}>\dfrac{-69}{144}\) nên \(\dfrac{-17}{36}>\dfrac{23}{-48}\) 

b) \(\dfrac{-1}{3}\) và \(\dfrac{2}{5}\) 

Vì \(\dfrac{-1}{3}\) là số âm mà \(\dfrac{2}{5}\) là số dương nên \(\dfrac{-1}{3}< \dfrac{2}{5}\) 

c) \(\dfrac{2}{7}\) và \(\dfrac{5}{4}\) 

Vì \(\dfrac{2}{7}< 1\) mà \(\dfrac{5}{4}>1\) nên \(\dfrac{2}{7}< \dfrac{5}{4}\) 

d) \(\dfrac{267}{-268}\) và \(\dfrac{-1347}{1343}\) 

\(\dfrac{267}{-268}=\dfrac{-267}{268}=\dfrac{-267.449}{268.449}=\dfrac{-119883}{120332}\) 

\(\dfrac{-1347}{1343}=\dfrac{-1347.89}{1343.89}=\dfrac{-119883}{119527}\) 

Vì \(\dfrac{-119883}{120332}>\dfrac{-119883}{119527}\) nên \(\dfrac{267}{-268}>\dfrac{-1347}{1343}\)

Bài 2:

\(\dfrac{5}{2}-\left(1\dfrac{3}{7}-0,4\right)=\dfrac{5}{2}-\dfrac{10}{7}-\dfrac{2}{5}=\dfrac{47}{70}\) 

Tính nhanh 5/8+5/24+5/48+......+5/9800