TMUA 2017 · Paper 1 · Question 11

The sequence $x_{n}$ is defined by the rules

x_{1} = 7

x_{n + 1} = \frac{23 x _{n} - 53}{5 x _{n} + 1}

The first three terms in the sequence are $7, 3, 1$

What is the value of $x_{100}$ ?