Periodic Recurrence Relation

Calculate the first few terms of the sequence:

• $x_0 = 1$
• $x_1 = 2$
• $x_2 = \frac{1 + x_1}{x_0} = \frac{1 + 2}{1} = 3$
• $x_3 = \frac{1 + x_2}{x_1} = \frac{1 + 3}{2} = 2$
• $x_4 = \frac{1 + x_3}{x_2} = \frac{1 + 2}{3} = 1$
• $x_5 = \frac{1 + x_4}{x_3} = \frac{1 + 1}{2} = 1$
• $x_6 = \frac{1 + x_5}{x_4} = \frac{1 + 1}{1} = 2$
• $x_7 = \frac{1 + x_6}{x_5} = \frac{1 + 2}{1} = 3$

The sequence is periodic with period 5: $(1, 2, 3, 2, 1, 1, 2, 3, 2, 1, \dots)$. Notice indices: $x_0 = 1, x_1 = 2, x_2 = 3, x_3 = 2, x_4 = 1$.

To find $x_{2021}$: $2021 \pmod 5 = 1$ So $x_{2021} = x_1 = 2$.