so sanh 2019\2021 + 2021\2019 voi 2
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NT
4
L
17 tháng 9 2019
Ta có:
2019.2021=2019.(2020+1)=2019.2020+2019 (1)
Lại có:
2020.2020=(2019+1).2020=2019.2020+2020 (2)
Vì 2019.2020=2019.2020 mà 2019<2020
=>(1)<(2)
=>.....
NT
7
27 tháng 3 2019
Ta có: \(\frac{a}{b}+\frac{b}{a}\le2\)
Dấu bằng xảy ra khi : a=b
=>\(\frac{2021}{2019}+\frac{2019}{2021}< 2\)
N
1
10 tháng 10 2020
Ta có: \(\sqrt{2020}-\sqrt{2019}=\frac{\left(\sqrt{2020}-\sqrt{2019}\right)\left(\sqrt{2020}+\sqrt{2019}\right)}{\sqrt{2020}+\sqrt{2019}}\)
\(=\frac{2020-2019}{\sqrt{2020}+\sqrt{2019}}=\frac{1}{\sqrt{2020}+\sqrt{2019}}\)
\(\sqrt{2021}-\sqrt{2020}=\frac{\left(\sqrt{2021}-\sqrt{2020}\right)\left(\sqrt{2021}+\sqrt{2020}\right)}{\sqrt{2021}+\sqrt{2020}}\)
\(=\frac{2021-2020}{\sqrt{2021}+\sqrt{2020}}=\frac{1}{\sqrt{2021}+\sqrt{2020}}\)
Vì \(\sqrt{2020}+\sqrt{2019}< \sqrt{2021}+\sqrt{2020}\)
\(\Rightarrow\) \(\frac{1}{\sqrt{2020}+\sqrt{2019}}>\frac{1}{\sqrt{2021}+\sqrt{2020}}\)
Hay \(\sqrt{2020}-\sqrt{2019}>\sqrt{2021}-\sqrt{2020}\)
Chúc bn học tốt!
4 tháng 10 2020
a) Ta có: \(\sqrt{2021}-\sqrt{2020}\)
\(=\frac{\left(\sqrt{2021}-\sqrt{2020}\right)\left(\sqrt{2021}+\sqrt{2020}\right)}{\sqrt{2021}+\sqrt{2020}}\)
\(=\frac{1}{\sqrt{2020}+\sqrt{2021}}\)
Ta có: \(\sqrt{2020}-\sqrt{2019}\)
\(=\frac{\left(\sqrt{2020}-\sqrt{2019}\right)\left(\sqrt{2020}+\sqrt{2019}\right)}{\sqrt{2020}+\sqrt{2019}}\)
\(=\frac{1}{\sqrt{2019}+\sqrt{2020}}\)
Ta có: \(\sqrt{2020}+\sqrt{2021}>\sqrt{2019}+\sqrt{2020}\)
\(\Leftrightarrow\frac{1}{\sqrt{2020}+\sqrt{2021}}< \frac{1}{\sqrt{2019}+\sqrt{2020}}\)
hay \(\sqrt{2021}-\sqrt{2020}< \sqrt{2020}-\sqrt{2019}\)
b) Ta có: \(\sqrt{2019\cdot2021}\)
\(=\sqrt{\left(2020-1\right)\left(2020+1\right)}\)
\(=\sqrt{2020^2-1}\)
Ta có: \(2020=\sqrt{2020^2}\)
Ta có: \(2020^2-1< 2020^2\)
nên \(\sqrt{2020^2-1}< \sqrt{2020^2}\)
\(\Leftrightarrow\sqrt{2019\cdot2021}< 2020\)
c) Ta có: \(\left(\sqrt{2019}+\sqrt{2021}\right)^2\)
\(=2019+2021+2\cdot\sqrt{2019\cdot2021}\)
\(=4040+2\sqrt{2019\cdot2021}\)
\(=4040+2\cdot\sqrt{2020^2-1}\)
Ta có: \(\left(2\sqrt{2020}\right)^2\)
\(=4\cdot2020\)
\(=4040+2\cdot2020\)
\(=4040+2\cdot\sqrt{2020^2}\)
Ta có: \(2020^2-1< 2020^2\)
\(\Leftrightarrow\sqrt{2020^2-1}< \sqrt{2020^2}\)
\(\Leftrightarrow2\cdot\sqrt{2020^2-1}< 2\cdot\sqrt{2020^2}\)
\(\Leftrightarrow4040+2\cdot\sqrt{2020^2-1}< 4040+2\cdot\sqrt{2020^2}\)
\(\Leftrightarrow\left(\sqrt{2019}+\sqrt{2021}\right)^2< \left(2\sqrt{2020}\right)^2\)
\(\Leftrightarrow\sqrt{2019}+\sqrt{2021}< 2\sqrt{2020}\)
17 tháng 4 2021
Giải:
Ta có: N=2019+2020/2020+2021
=>N=2019/2020+2021 + 2020/2020+2021
Vì 2019/2020 > 2019/2020+2021 ; 2020/2021 > 2020/2020+2021
=>M>N
Vậy ...
Chúc bạn học tốt!
17 tháng 4 2021
Ta có : \(\dfrac{2019}{2020}>\dfrac{2019}{2020+2021}\)
\(\dfrac{2020}{2021}>\dfrac{2020}{2020+2021}\)
\(\Rightarrow\dfrac{2019}{2020}+\dfrac{2020}{2021}>\dfrac{2019+2020}{2020+2021}\)
\(\Rightarrow M>N\)
AH
Akai Haruma
Giáo viên
12 tháng 2 2023
Lời giải:
$A=1-\frac{1}{2019}+1-\frac{1}{2020}+1-\frac{1}{2021}+1+\frac{3}{2018}$
$=4+(\frac{1}{2018}-\frac{1}{2019}+\frac{1}{2018}-\frac{1}{2020}+\frac{1}{2018}-\frac{1}{2021})$
$> 4+0+0+0+0=4$
Ta có:
\(\frac{2019}{2021}< 1\)
\(\frac{2021}{2019}>1\)
\(\Rightarrow\frac{2019}{2021}+\frac{2021}{2019}< 2\)