A coin with the probability of coming up heads equal to $p$ is tossed $n$ times. Let $a _ { n }$ be the probability that the coin never comes up heads twice in a row, and let $b _ { n }$ be the probability that the coin never comes up heads three times in a row. Here, let $a _ { 1 } = 1$ and $b _ { 1 } = b _ { 2 } = 1$. Answer the following questions.
(1) Obtain $a _ { 2 }$ as a function of $p$.
(2) When $n \geq 3$, describe $a _ { n }$ by using $p , a _ { n - 1 }$ and $a _ { n - 2 }$.
(3) When $p = \frac { 2 } { 3 }$, obtain all the pairs $( \alpha , \beta )$ of real numbers that satisfy the following recursion:
$$a _ { n } + \alpha a _ { n - 1 } = \beta \left( a _ { n - 1 } + \alpha a _ { n - 2 } \right)$$
(4) When $p = \frac { 2 } { 3 }$, obtain $a _ { n }$ as a function of $n$.
(5) Obtain $b _ { 3 }$ as a function of $p$.
(6) When $n \geq 4$, describe $b _ { n }$ by using $p , b _ { n - 1 } , b _ { n - 2 }$ and $b _ { n - 3 }$.
(7) When $p = \frac { 3 } { 4 }$, show that the following equation holds for any positive integer $n$:
$$\begin{aligned} b _ { n } = & \frac { 9 } { 8 } \left( \frac { 3 } { 4 } \right) ^ { n - 1 } - \frac { ( - 1 ) ^ { n - 1 } } { 8 } \left( \frac { \sqrt { 3 } } { 4 } \right) ^ { n - 1 } \cos ( ( n - 1 ) \theta ) \\ & - \frac { \sqrt { 2 } \mathrm { i } } { 8 } \left\{ \left( \frac { - 1 + \sqrt { 2 } \mathrm { i } } { 4 } \right) ^ { n - 1 } - \left( \frac { - 1 - \sqrt { 2 } \mathrm { i } } { 4 } \right) ^ { n - 1 } \right\} \end{aligned}$$
where i is the imaginary unit and $\theta$ is the angle that satisfies $\cos \theta = \frac { 1 } { \sqrt { 3 } }$ and $\sin \theta = \frac { \sqrt { 2 } } { \sqrt { 3 } }$.