There is a real number sequence $\left\langle a_{n} \right\rangle$, where $a_{n} = \cos\left(n\pi - \frac{\pi}{6}\right)$, and $n$ is a positive integer. Select the correct options.
(1) $a_{1} = -\frac{1}{2}$
(2) $a_{2} = a_{3}$
(3) $a_{4} = a_{24}$
(4) $\left\langle a_{n} \right\rangle$ is a convergent sequence, and $\lim_{n \rightarrow \infty} a_{n} < 1$
(5) $\sum_{n=1}^{\infty} \left(a_{n}\right)^{n} = 3 - 2\sqrt{3}$

A sequence $< a _ { n } >$ satisfies $3 a _ { n + 1 } = a _ { n } + n$ (for all positive integers $n$) and $a _ { 1 } = 2$. Let the sequence $< b _ { n } >$ satisfy $b _ { n } = a _ { n } - \frac { n } { 2 } + \frac { 3 } { 4 }$. Select the correct options.
(1) $a _ { 2 } = 2$
(2) $b _ { 2 } = \frac { 3 } { 4 }$
(3) The sequence $< b _ { n } >$ is a geometric sequence with common ratio $\frac { 2 } { 3 }$
(4) For any positive integer $n$, $3 ^ { n } a _ { n }$ is always a positive integer
(5) $b _ { 10 } < 10 ^ { - 4 }$

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 $\sin \theta = \frac { \sqrt { 2 } } { \sqrt { 3 } }$.

The function f satisfies the equation
$$f ( n ) = 2 \cdot f ( n - 1 ) + 1$$
for integers $n \geq 1$. Given that $f ( 0 ) = 1$, what is $f ( 2 )$?
A) 8
B) 7
C) 6
D) 5
E) 4

The sequence $(a _ { k })$ is defined as
$$\begin{aligned} & a _ { 1 } = 40 \\ & a _ { k + 1 } = a _ { k } - k \quad ( k = 1,2,3 , \ldots ) \end{aligned}$$
Accordingly, what is the term $\mathrm { a } _ { 8 }$?
A) 4
B) 7
C) 12
D) 15
E) 19

Let $a _ { 1 } , a _ { 2 }$ be real numbers. The sequence $\left( a _ { n } \right)$ satisfies the relation
$$a _ { n + 2 } = a _ { n + 1 } + a _ { n } \quad ( n = 1,2 , \cdots )$$
Given that $a _ { 8 } = 6$, what is the sum $a _ { 6 } + a _ { 9 }$?
A) 9
B) 10
C) 12
D) 15
E) 16

For a sequence $a_n$ where the sum of any three consecutive terms is equal to each other,
$$a _ { 2 } + a _ { 3 } = a _ { 4 } = 2$$
equality is satisfied.
Accordingly, $$a _ { 1 } + a _ { 2 } + \ldots + a _ { 25 }$$
what is the result of the sum?
A) 34
B) 35
C) 36
D) 37
E) 38

The sequence $(a_n)$ of real numbers satisfies for every positive integer $n$
$$a_{n+1} = a_n + \frac{(-1)^n \cdot a_n}{2}$$
the equality. If $a_5 = 18$, what is $a_1$?
A) 4
B) 8
C) 16
D) 32
E) 64

The sequence $(a_n)$ of real numbers satisfies for every positive integer $n$
$$a_{n} + (-1)^{n} \cdot a_{n+1} = 2^{n}$$
If $a_{1} = 0$, what is the sum $a_{3} + a_{4} + a_{5} + a_{6}$?
A) 6 B) 8 C) 12 D) 16 E) 20

Selma designs a toy with identical clips in yellow and blue colors. In step 1, she places one clip on the ground. In each subsequent step, she attaches one clip to each blue part of all the clips she placed in the previous step as shown in the figure, and moves to the next step. Selma completes the first 3 steps of this toy using 7 clips.
Accordingly, after Selma completes step 12, how many more clips has she used in total compared to after completing step 10?
A) $3 \cdot 2^{10}$
B) $3 \cdot 2^{11}$
C) $7 \cdot 2^{9}$
D) $7 \cdot 2^{10}$
E) $2^{11}$