Tham khảo:

Sửa đề: So sánh Q và 5/16

Ta có: \(Q=\frac15+\frac{2}{5^2}+\frac{3}{5^3}+\cdots+\frac{2022}{5^{2022}}\)

=>\(5Q=1+\frac25+\frac{3}{5^2}+\cdots+\frac{2022}{5^{2021}}\)

=>\(5Q-Q=1+\frac25+\frac{3}{5^2}+\cdots+\frac{2022}{5^{2021}}-\frac15-\frac{2}{5^2}-\cdots-\frac{2022}{5^{2022}}\)

=>\(4Q=1+\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2021}}-\frac{2022}{5^{2022}}\)

Đặt \(A=\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2021}}\)

=>\(5A=1+\frac15+\cdots+\frac{1}{5^{2020}}\)

=>\(5A-A=1+\frac15+\cdots+\frac{1}{5^{2020}}-\frac15-\frac{1}{5^2}-\cdots-\frac{1}{5^{2021}}\)

=>\(4A=1-\frac{1}{5^{2021}}=\frac{5^{2021}-1}{5^{2021}}\)

=>\(A=\frac{5^{2021}-1}{4\cdot5^{2021}}\)

Ta có: \(4Q=1+\frac15+\frac{1}{5^2}+\cdots+\frac{1}{5^{2021}}-\frac{2022}{5^{2022}}\)

\(=1+\frac{5^{2021}-1}{4\cdot5^{2021}}-\frac{2022}{5^{2022}}=1+\frac{5^{2022}-5-8088}{4\cdot5^{2022}}=1+\frac{5^{2022}-8093}{4\cdot5^{2022}}\)

=>\(4Q=1+\frac14-\frac{8093}{4\cdot5^{2022}}<\frac54\)

=>\(Q<\frac{5}{16}\)

Lời giải:

Ta thấy: $\frac{2021^2+1}{2021}=2021+\frac{1}{2021}< 2022< 2022+\frac{1}{2022}=\frac{2022^2+1}{2022}$

$\Rightarrow \frac{2021}{2021^2+1}> \frac{2022}{2022^2+1}$

\(\dfrac{1}{10}A=\dfrac{10^{2023}+5}{10^{2023}+50}=1-\dfrac{45}{10^{2023}+50}\)

\(\dfrac{1}{10}B=\dfrac{10^{2022}+5}{10^{2022}+50}=1-\dfrac{45}{10^{2022}+50}\)

10^2023+50>10^2022+50

=>-45/10^2023+50<-45/10^2020+50

=>1/10A<1/10B

=>A<B

A phải lớn hơn B vì phần bù của số nào nhỏ hơn thì số đó lớn hơn bạn nhé. Nhưng dù sao cx động viên bạn, mình tick cho. Cảm ơn bạn nhiều

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:

\(2023^{2022}=2023\cdot2023^{2021}\)

\(2022^{2022}+2022^{2021}=2022^{2021}\cdot\left(2022+1\right)=2023\cdot2022^{2021}\)

Mà: \(2023>2022\)

\(\Rightarrow2023^{2021}>2022^{2021}\)

\(\Rightarrow2023^{2021}\cdot2023>2022^{2021}\cdot2023\)

\(\Rightarrow2023^{2022}>2022^{2022}+2022^{2021}\) 

Vậy: ...