SO SÁNH:
A = \(\frac{2021^{2021}+1}{2021^{2022}+1}\)và B = \(\frac{2021^{2022}-1}{2021^{2023}-1}\)
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Những câu hỏi liên quan
3 tháng 3 2023
a: \(98^{10}\cdot A=\dfrac{98^{98}+98^{10}}{98^{98}+1}=1+\dfrac{98^{10}-1}{98^{98}+1}\)
\(98^{10}\cdot B=\dfrac{98^{99}+98^{10}}{98^{99}+1}=1+\dfrac{98^{10}-1}{98^{99}+1}\)
98^88+1>98^99+1
=>A<B
b: \(\dfrac{1}{2022^2}\cdot C=\dfrac{2022^{2023}+1}{2022^{2023}+2022^2}=1+\dfrac{1-2022^2}{2022^{2023}+2022^2}\)
\(\dfrac{1}{2022^2}\cdot D=\dfrac{2022^{2021}+1}{2022^{2021}+2022^2}=1+\dfrac{1-2022^2}{2022^{2021}+2022^2}\)
2022^2023>2022^2021
=>2022^2023+2022^2>2022^2021+2022^2
=>\(\dfrac{2022^2-1}{2022^{2023}+2022^2}< \dfrac{2022^2-1}{2022^{2021}+2022^2}\)
=>\(\dfrac{1-2022^2}{2022^{2023}+2022^2}>\dfrac{1-2022^2}{2022^{2021}+2022^2}\)
=>C>D
12 tháng 3 2023
\(10A=\dfrac{10^{2023}+10}{10^{2023}+1}=1+\dfrac{9}{10^{2023}+1}\)
\(10B=\dfrac{10^{2022}+10}{10^{2022}+1}=1+\dfrac{9}{10^{2022}+1}\)
2023>2022
=>10^2023+1>10^2022+1
=>10A<10B
=>A<B
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
1 tháng 11
A = \(\dfrac{1}{2021.2022}\) + \(\dfrac{1}{2022.2023}\) + \(\dfrac{1}{2023.2024}\) + \(\dfrac{1}{2024.2025}\) - \(\dfrac{4}{2021.2025}\)
A = \(\dfrac{1}{2021}\) - \(\dfrac{1}{2022}\) + \(\dfrac{1}{2022}\) - \(\dfrac{1}{2023}\) + \(\dfrac{1}{2023}\) - \(\dfrac{1}{2024}\) + \(\dfrac{1}{2024}\) - \(\dfrac{1}{2025}\) - \(\dfrac{1}{2021}\) + \(\dfrac{1}{2025}\)
A = (\(\dfrac{1}{2021}\) - \(\dfrac{1}{2021}\)) + (\(\dfrac{1}{2022}\) - \(\dfrac{1}{2022}\)) + (\(\dfrac{1}{2023}\) - \(\dfrac{1}{2023}\)) + (\(\dfrac{1}{2024}\) - \(\dfrac{1}{2024}\)) + (\(\dfrac{1}{2025}\) - \(\dfrac{1}{2025}\))
A = 0 + 0 +0 + 0+ ... + 0
A = 0
MM
1
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
3 tháng 5 2023
B = \(\dfrac{1}{2002}\) + \(\dfrac{2}{2021}\) + \(\dfrac{3}{2020}\)+...+ \(\dfrac{2021}{2}\) + \(\dfrac{2022}{1}\)
B = \(\dfrac{1}{2002}\) + \(\dfrac{2}{2021}\) + \(\dfrac{3}{2020}\)+...+ \(\dfrac{2021}{2}\) + 2022
B = 1 + ( 1 + \(\dfrac{1}{2022}\)) + ( 1 + \(\dfrac{2}{2021}\)) + \(\left(1+\dfrac{3}{2020}\right)\)+ ... + \(\left(1+\dfrac{2021}{2}\right)\)
B = \(\dfrac{2023}{2023}\) + \(\dfrac{2023}{2022}\) + \(\dfrac{2023}{2021}\) + \(\dfrac{2023}{2020}\) + ...+ \(\dfrac{2023}{2}\)
B = 2023 \(\times\) ( \(\dfrac{1}{2023}\) + \(\dfrac{1}{2022}\) + \(\dfrac{1}{2021}\) + \(\dfrac{1}{2020}\)+ ... + \(\dfrac{1}{2}\))
Vậy B > C
DD
Đoàn Đức Hà
Giáo viên
18 tháng 3 2022
Ta có: \(\frac{2022}{2021^2+k}\le\frac{2022}{2021^2}\) (với \(k\)là số tự nhiên bất kì)
Ta có:
\(A=\frac{2022}{2021^2+1}+\frac{2022}{2021^2+2}+...+\frac{2022}{2021^2+2021}\)
\(\le\frac{2022}{2021^2}+\frac{2022}{2021^2}+...+\frac{2022}{2021^2}=\frac{2022}{2021^2}.2021=\frac{2022}{2021}\)
Ta có: \(\frac{2022}{2021^2+k}>\frac{2022}{2021^2+2021}=\frac{2022}{2021.2022}=\frac{1}{2021}\)với \(k\)tự nhiên, \(k< 2021\))
Suy ra \(A=\frac{2022}{2021^2+1}+\frac{2022}{2021^2+2}+...+\frac{2022}{2021^2+2021}\)
\(>\frac{1}{2021}+\frac{1}{2021}+...+\frac{1}{2021}=\frac{2021}{2021}=1\)
Suy ra \(1< A\le\frac{2022}{2021}\)do đó \(A\)không phải là số tự nhiên.
PA
2
23 tháng 8 2021
Nhỏ hơn
Ta có 2020/2021 <1
2021/2022 <1
2022/2023 <1
2023/2024 <1
Suy ra A=(2021/2021+2021/2022 +2022/2023 +2023/2024) < (1+1+1+1)= 4
Vậy A <4
Chúc bạn học tốt
23 tháng 8 2021
\(\dfrac{2020}{2021}< 1\)
\(\dfrac{2021}{2022}< 1\)
\(\dfrac{2021}{2022}< 1\)
\(\dfrac{2023}{2024}< 1\)
Do đó: A<4
24 tháng 7 2023
a: \(B=\dfrac{154}{155+156}+\dfrac{155}{155+156}\)
\(\dfrac{154}{155}>\dfrac{154}{155+156}\)
\(\dfrac{155}{156}>\dfrac{155}{155+156}\)
=>154/155+155/156>(154+155)/(155+156)
=>A>B
b: \(C=\dfrac{2021+2022+2023}{2022+2023+2024}=\dfrac{2021}{6069}+\dfrac{2022}{6069}+\dfrac{2023}{6069}\)
2021/2022>2021/6069
2022/2023>2022/2069
2023/2024>2023/6069
=>D>C
Ta có:
\(A=\frac{2021^{2021}+1}{2021^{2022}+1}\Leftrightarrow10A=\frac{2021^{2022}+10}{2021^{2022}+1}=1+\frac{9}{2021^{2022}+1}\)
\(B=\frac{2021^{2022}-1}{2021^{2023}-1}\Leftrightarrow10B=\frac{2021^{2023}-10}{2021^{2023}-1}=1-\frac{9}{2021^{2023}-1}\)
Hay ta đang so sánh: \(\frac{9}{2021^{2022}};\frac{9}{2021^{2023}}\)
Mà \(\frac{9}{2021^{2022}}>\frac{9}{2021^{2023}}\)nên \(\frac{2021^{2021}+1}{2021^{2022}+1}>\frac{2021^{2022}-1}{2021^{2023}-1}\)hay\(A>B\)
Vậy \(A>B\)
Cảm ơn bạn Nguyễn Đăng Nhân nha !!!