\(a,ĐKXĐ:x-1\ge0\Leftrightarrow x\ge1\)

Đặt \(\hept{\begin{cases}\sqrt[3]{2-x}=a\\\sqrt{x-1}=b\left(b\ge0\right)\end{cases}\Rightarrow}a^3+b^2=2-x+x-1=1\)

Lại có: \(a=1-b\)

Thay vào được

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

\(\Leftrightarrow1-3b+3b^2-b^3+b^2-1=0\)

\(\Leftrightarrow-b^3+4b^2-3b=0\)

\(\Leftrightarrow b^3-4b^2+3b=0\)

\(\Leftrightarrow b\left(b^2-4b+3\right)=0\)

\(\Leftrightarrow b\left(b-1\right)\left(b-3\right)=0\)

\(\Leftrightarrow b=0\left(h\right)b=1\left(h\right)b=3\)(T/m ĐK b>0)

*Với b = 0

\(\Leftrightarrow\sqrt{x-1}=0\)

\(\Leftrightarrow x=1\left(TmĐKXĐ\right)\)

*Với b = 1

\(\Leftrightarrow\sqrt{x-1}=1\)

\(\Leftrightarrow x-1=1\)

\(\Leftrightarrow x=2\left(TmĐKXĐ\right)\)

*Với b = 3

\(\Leftrightarrow\sqrt{x-1}=3\)

\(\Leftrightarrow x-1=9\)

\(\Leftrightarrow x=10\)

Vậy \(S\in\left\{1;2;10\right\}\)

em chỉ bt bài 2 nha!

\(A=\left(1-\frac{2}{2\cdot3}\right)\left(1-\frac{2}{3\cdot4}\right)...\left(1-\frac{2}{2020\cdot2021}\right)\)

\(\frac{2}{3}\cdot\frac{5}{6}\cdot\frac{9}{10}\cdot...\cdot\frac{2020\cdot2021-2}{2020\cdot2021}\left(1\right)\)

Mặt khác:\(2020\cdot2021-2=2020\left(2022-1\right)+2020-2022\)

\(=2020\cdot2022-2022\)

\(=2022\left(2020-1\right)=2019\cdot2022\left(2\right)\)

Từ (1),(2) ta có:

\(A=\frac{4\cdot1}{2\cdot3}\cdot\frac{5\cdot2}{3\cdot4}\cdot...\cdot\frac{2022\cdot2019}{2020\cdot2021}\)

\(=\frac{\left(4\cdot5\cdot6\cdot...\cdot2022\right)\left(1\cdot2\cdot3\cdot...\cdot2019\right)}{\left(2\cdot3\cdot4\cdot...\cdot2020\right)\left(3\cdot4\cdot5\cdot...\cdot2021\right)}\)

\(=\frac{2021\cdot2022}{2\cdot3}\cdot\frac{1\cdot2}{2020\cdot2021}=\frac{2022}{3\cdot2020}=\frac{2022}{6060}\)

Ta có: \(A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{2019\cdot2020}+\dfrac{1}{2020\cdot2021}\)

\(=\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2019}-\dfrac{1}{2020}+\dfrac{1}{2020}-\dfrac{1}{2021}\)

\(=\dfrac{1}{1}-\dfrac{1}{2021}=\dfrac{2021}{2021}-\dfrac{1}{2021}\)

\(=\dfrac{2020}{2021}\)

mà \(\dfrac{2020}{2021}< \dfrac{2021}{2021}=1\)

nên A<1

Ta có: \(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{2020\cdot2021}+\dfrac{1}{2021\cdot2022}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2020}-\dfrac{1}{2021}+\dfrac{1}{2021}-\dfrac{1}{2022}\)

\(=1-\dfrac{1}{2022}=\dfrac{2021}{2022}\)

1/1x2+1/2x3+1/3x4+...+1/2020x2021+1/2021x2022

=1/1-1/2+1/2-1/3+1/3-1/4+...+1/2020-1/2021+1/2021-1/2022.

=1/1-1/2022

=2021/2022

S = 4/2.3 . 10/3.4 ..........9898/ 99.100

S= 1.4/2.3 . 2.5/3.4 . 3.6/4.5 .......98.101/99.100

S = (1.2.3...98).(4.5.6...100).1014/(2.3.4...98).99.(4.5.6...100).3

S=101/99.3

S=101.297