Trả lời:

\(P=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{2021^2}-1\right)\)

\(=\frac{1-2^2}{2^2}\cdot\frac{1-3^2}{3^2}\cdot\frac{1-4^2}{4^2}\cdot...\cdot\frac{1-2021^2}{2021^2}\)

\(=\frac{-3}{2^2}\cdot\frac{-8}{3^2}\cdot\frac{-15}{4^2}\cdot...\cdot\frac{-4084440}{2021^2}\)

\(=\frac{3}{2^2}\cdot\frac{8}{3^2}\cdot\frac{15}{4^2}\cdot...\cdot\frac{4084440}{2021^2}\) ( vì tích trên có 2020 thừa số, mà tích của 2020 thừa số âm là số dương )

\(=\frac{3\cdot8\cdot15\cdot...\cdot4084440}{2^2\cdot3^2\cdot4^2\cdot...\cdot2021^2}\)

\(=\frac{1\cdot3\cdot2\cdot4\cdot3\cdot5\cdot...\cdot2020\cdot2022}{2\cdot2\cdot3\cdot3\cdot4\cdot4\cdot...\cdot2021\cdot2021}\)

\(=\frac{\left(1\cdot2\cdot3\cdot...\cdot2020\right)\cdot\left(3\cdot4\cdot5\cdot...\cdot2022\right)}{\left(2\cdot3\cdot4\cdot...\cdot2021\right)\cdot\left(2\cdot3\cdot4\cdot...\cdot2021\right)}\)

\(=\frac{1\cdot2022}{2021\cdot2}=\frac{1011}{2021}>\frac{1011}{2022}=\frac{1}{2}\)

Vậy \(P>\frac{1}{2}\)

Kết luận B<1/2

\(P=\left(\frac{1}{2^2}-1\right)\times\left(\frac{1}{3^2}-1\right)\times...\times\left(\frac{1}{2021^2}-1\right)\)

\(=\left(1-\frac{1}{2^2}\right)\times\left(1-\frac{1}{3^2}\right)\times...\times\left(1-\frac{1}{2021^2}\right)\)

\(=\frac{2^2-1}{2^2}\times\frac{3^2-1}{3^2}\times...\times\frac{2021^2-1}{2021^2}\)

\(=\frac{1\times3}{2^2}\times\frac{2\times4}{3^2}\times\frac{3\times5}{4^2}\times...\times\frac{2020\times2022}{2021^2}\)

\(=\frac{\left(1\times2\times3\times...\times2020\right)\times\left(3\times4\times5\times...\times2022\right)}{\left(2\times3\times4\times...\times2021\right)\times\left(2\times3\times4\times...\times2021\right)}\)

\(=\frac{1\times2022}{2021\times2}=\frac{2022}{2021\times2}>\frac{2021}{2021\times2}=\frac{1}{2}\)

Đây dang lớp mấy vây

sử dụng phương pháp miền giá trị

bạn nói rõ hơn được không?

Xét phân thức phụ sau, với n nguyên dương lớn hơn 1 ta có:

Ta có: \(\frac{1}{\left(n+1\right)\sqrt{n}}=\frac{\left(n+1\right)-n}{\left(n+1\right)\sqrt{n}}=\frac{\left(\sqrt{n+1}-\sqrt{n}\right)\left(\sqrt{n+1}+\sqrt{n}\right)}{\left(n+1\right)\sqrt{n}}\)

\(< \frac{2\sqrt{n+1}\left(\sqrt{n+1}-\sqrt{n}\right)}{\left(\sqrt{n+1}\right)^2\sqrt{n}}=2\left(\frac{\sqrt{n+1}-\sqrt{n}}{\left(\sqrt{n+1}\right)\sqrt{n}}\right)\)

\(=2\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\)

=> \(\frac{1}{\left(n+1\right)\sqrt{n}}< 2\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\)

Áp dụng vào bài toán ta được:

\(A=2\left(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2019}}-\frac{1}{\sqrt{2020}}\right)\)

\(A=2-\frac{2}{\sqrt{2020}}< 2=B\)

Vậy A < B

lớp 8a3 nguyễn khuyến đúng ko

Bài 1 :

\(M=\dfrac{30-2^{20}}{2^{18}}=\dfrac{2.15-2^{20}}{2^{18}}=\dfrac{15}{2^{17}}-2^2=\dfrac{15}{2^{17}}-4< 0\left(\dfrac{15}{2^{17}}< 1\right)\)

\(N=\dfrac{3^5}{1^{2021}+2^3}=\dfrac{3^5}{9}=\dfrac{3^5}{3^2}=3^3=27\)

\(\Rightarrow M< N\)

Bài 3 :

a) \(t^2+5t-8\) khi \(t=2\)

\(=5^2+2.5-8\)

\(=25+10-8\)

\(=27\)

b) \(\left(a+b\right)^2-\left(b-a\right)^3+2021\left(1\right)\)

\(\left\{{}\begin{matrix}a=5\\b=a+1=6\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a+b=11\\b-a=1\end{matrix}\right.\)

\(\left(1\right)=11^2-1^3+2021=121-1+2021=2141\)

c) \(x^3-3x^2y+3xy^2-y^3=\left(x-y\right)^3\left(1\right)\)

\(\left\{{}\begin{matrix}x=3\\y=2\end{matrix}\right.\) \(\Rightarrow x-y=1\)

\(\left(1\right)=1^3=1\)

Bài 2 :

a) \(...=2^3\left(19-14\right)+1=8.5+1=41\)

b) \(...=100-\left[60:\left(5^2-15\right)\right]=100-\left[60:10\right]=100-6=94\)

c) \(...=160:\left[17+\left(9.5-\left(14+2^3\right)\right)\right]=160:\left[17+\left(45-22\right)\right]=160:\left[17+23\right]=160:40=4\)

d) \(...=798+100\left[16-2\left(25-22\right)\right]=798+100\left[16-2.3\right]=798+100.10=798+1000=1798\)