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Given that, $x_{0}=1, x_{1}=2, x_{n+2} = \frac{1+x_{n+1}}{x_{n}}; n=0,1,2,3,\dots$

Now, 

  • $x_{0}=1$
  • $x_{1}=2$
  • $x_{2}=\frac{1+2}{1} = 3$
  • $x_{3}=\frac{1+3}{2} = 2$
  • $x_{4}=\frac{1+2}{3} = 1$
  • $x_{5}=\frac{1+1}{2} = 1$
  • $x_{6}=\frac{1+1}{1} = 2$
  • $x_{7}=\frac{1+2}{1} = 3$
  • $x_{8}=\frac{1+3}{2} = 2$
  • $x_{9}=\frac{1+2}{3} = 1$
  • $\vdots \quad \vdots \quad \vdots \quad \vdots$

We can see the pattern,  $1,2,3,2,{\color{Red} {1}},1,2,3,2,{\color{Red} {1}},\dots$

Every $5^{\text{th}}$ multiple is $1.$

So, $x_{2020}=1.$

$\therefore$ The value of $x_{2021} = 2.$

Correct Answer $:\text{D}$

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