Question 40

A sequence $$\left\{x_n \right\}$$ of real numbers is defined as follows:

$$x_0 = 1, x_1 = 2,$$ and $$x_n = \frac{1 + x_{n - 1}}{x_{n - 2}}$$ for n = 2, 3, 4 ...
It follows that $$x_{2018}$$ is

Solution

Given that $$x_0=1\ and\ x_1=2\ $$

From above formula we get $$x_2=3,\ x_3=2,\ x_4=1,\ x_5\ =\ 1\ and\ x_6=2$$

We see that the pattern repeats itself in the period of five with $$x_{5k}=1,\ x_{5k+1}=2,\ x_{5k+2}=3,\ x_{5k+3}\ =\ 2\ and\ x_{5k+4}=1$$

2018 is of form 5k+3 hence $$x_{2018}=2$$

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