# Matt's Recurrence

Geometry Level 5

Let $$\theta = \sin ^{-1} \frac {7}{25}$$. Consider the sequence of values defined by $$a_n = \sin ( n \theta)$$. They satisfy the recurrence relation $a_{n+2} = k_1 a_{n+1} + k_0 a_{ n}, \quad n \in \mathbb{N}$ for some (fixed) real numbers $$k_1 , k_0$$. The sum $$k_1 + k_0$$ can be written as $$\frac {p}{q}$$, where $$p$$ and $$q$$ are positive coprime integers. What is the value of $$p + q$$?

This problem is proposed by Matt.

Details and assumptions

By definition, $$\sin ^{-1} : [-1 ,1 ] \rightarrow [ - \frac {\pi}{2} , \frac {\pi}{2} ]$$.

×