We define, for every natural integer $n$, the complex numbers $z$ by:
$$\begin{cases} z_{0} & = 16 \\ z_{n+1} & = \frac{1 + \mathrm{i}}{2} z_{n}, \text{ for every natural integer } n. \end{cases}$$
We denote $r_{n}$ the modulus of the complex number $z_{n}$: $r_{n} = |z_{n}|$. In the plane equipped with a direct orthonormal coordinate system with origin O, we consider the points $A_{n}$ with affixes $z_{n}$.

1. a) Calculate $z_{1}, z_{2}$ and $z_{3}$. b) Plot the points $A_{1}$ and $A_{2}$ on the graph given in the appendix, to be returned with your answer sheet. c) Write the complex number $\frac{1 + \mathrm{i}}{2}$ in trigonometric form. d) Prove that the triangle $\mathrm{OA}_{0}A_{1}$ is isosceles right-angled at $A_{1}$.
2. Prove that the sequence $(r_{n})$ is geometric, with common ratio $\frac{\sqrt{2}}{2}$.

Is the sequence $(r_{n})$ convergent? Interpret the previous result geometrically. We denote $L_{n}$ the length of the broken line connecting point $A_{0}$ to point $A_{n}$ passing successively through points $A_{1}, A_{2}, A_{3}$, etc. Thus $L_{n} = \sum_{i=0}^{n-1} A_{i}A_{i+1} = A_{0}A_{1} + A_{1}A_{2} + \ldots + A_{n-1}A_{n}$.
3. a) Prove that for every natural integer $n$: $A_{n}A_{n+1} = r_{n+1}$. b) Give an expression for $L_{n}$ as a function of $n$. c) Determine the possible limit of the sequence $(L_{n})$.

For a constant $k$ with $k > 1$, there is a sequence $\left\{ a _ { n } \right\}$ satisfying the following conditions.
For all natural numbers $n$, $a _ { n } < a _ { n + 1 }$ and the slope of the line passing through two points $\mathrm { P } _ { n } \left( a _ { n } , 2 ^ { a _ { n } } \right)$ and $\mathrm { P } _ { n + 1 } \left( a _ { n + 1 } , 2 ^ { a _ { n + 1 } } \right)$ on the curve $y = 2 ^ { x }$ is $k \times 2 ^ { a _ { n } }$.
Let $\mathrm { Q } _ { n }$ be the point where the line passing through $\mathrm { P } _ { n }$ parallel to the $x$-axis and the line passing through $\mathrm { P } _ { n + 1 }$ parallel to the $y$-axis meet, and let $A _ { n }$ be the area of triangle $\mathrm { P } _ { n } \mathrm { Q } _ { n } \mathrm { P } _ { n + 1 }$. The following is the process of finding $A _ { n }$ when $a _ { 1 } = 1$ and $\frac { A _ { 3 } } { A _ { 1 } } = 16$.
Since the slope of the line passing through two points $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ is $k \times 2 ^ { a _ { n } }$, $$2 ^ { a _ { n + 1 } - a _ { n } } = k \left( a _ { n + 1 } - a _ { n } \right) + 1$$ Thus, for all natural numbers $n$, $a _ { n + 1 } - a _ { n }$ is a solution of the equation $2 ^ { x } = k x + 1$. Since $k > 1$, the equation $2 ^ { x } = k x + 1$ has exactly one positive real root $d$. Therefore, for all natural numbers $n$, $a _ { n + 1 } - a _ { n } = d$, and the sequence $\left\{ a _ { n } \right\}$ is an arithmetic sequence with common difference $d$. Since the coordinates of point $\mathrm { Q } _ { n }$ are $\left( a _ { n + 1 } , 2 ^ { a _ { n } } \right)$, $$A _ { n } = \frac { 1 } { 2 } \left( a _ { n + 1 } - a _ { n } \right) \left( 2 ^ { a _ { n + 1 } } - 2 ^ { a _ { n } } \right)$$ Since $\frac { A _ { 3 } } { A _ { 1 } } = 16$, the value of $d$ is (가), and the general term of the sequence $\left\{ a _ { n } \right\}$ is $$a _ { n } = \text { (나) }$$ Therefore, for all natural numbers $n$, $A _ { n } =$ (다).
When the number corresponding to (가) is $p$, and the expressions corresponding to (나) and (다) are $f ( n )$ and $g ( n )$ respectively, what is the value of $p + \frac { g ( 4 ) } { f ( 2 ) }$? [4 points]
(1) 118
(2) 121
(3) 124
(4) 127
(5) 130

Given a sequence $\{ a _ { n } \}$ satisfying $a _ { 1 } = 1$ and $n a _ { n - 1 } = 2 ( n + 1 ) a _ { n }$. Let $b _ { n } = \frac { a _ { n } } { n }$.
(1) Find $b _ { 1 } , b _ { 2 } , b _ { 3 }$;
(2) Determine whether the sequence $\{ b _ { n } \}$ is a geometric sequence and explain the reasoning;
(3) Find the general term formula for $\{ a _ { n } \}$.