20. (This question is worth 13 points) Given the sequence $\left\{ a _ { n } \right\}$ satisfying: $a _ { 1 } \in \mathbf { N } ^ { * } , a _ { 1 } \leqslant 36$, and $a _ { n + 1 } = \left\{ \begin{array} { l } 2 a _ { n } , a _ { n } \leqslant 18 , \\ 2 a _ { n } - 36 , a _ { n } > 18 \end{array} ( n = 1,2 , \ldots ) \right.$. Let the set $M = \left\{ a _ { n } \mid n \in \mathbf { N } ^ { * } \right\}$. (I) If $a _ { 1

20. (This question is worth 15 points) Given that the sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = \frac { 1 } { 2 }$ and $a _ { n + 1 } = a _ { n } - a _ { n } ^ { 2 }$ ( $n \in \mathbb{N

Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. If $S _ { n } = 2 a _ { n } + 1$, then $S _ { 6 } = \_\_\_\_$

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 $\{ a _ { n } \}$.

17. (12 points)
A sequence $\left\{ a _ { n } \right\}$ satisfies $\frac { 1 } { a _ { n + 1 } } - \frac { 2 } { a _ { n } } = 0$, and $a _ { 1 } = \frac { 1 } { 2 }$.
(1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$;
(2) Find the sum $S _ { n }$ of the first $n$ terms of the sequence $\left\{ \frac { 1 } { a _ { n } } + 2 n \right\}$.

19. (12 points)
Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 1 , b _ { 1 } = 0,4 a _ { n + 1 } = 3 a _ { n } - b _ { n } + 4,4 b _ { n + 1 } = 3 b _ { n } - a _ { n } - 4$.
(1) Prove that $\left\{ a _ { n } + b _ { n } \right\}$ is a geometric sequence and $\left\{ a _ { n } - b _ { n } \right\}$ is an arithmetic sequence Therefore, the polar equation of the locus of point $P$ is $\rho = 4 \cos \theta , \theta \in \left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$ .

Given a sequence $\left\{ a _ { n } \right\}$, if we select the $i _ { 1 }$-th term, the $i _ { 2 }$-th term, $\cdots$, the $i _ { m }$-th term $\left( i _ { 1 } < i _ { 2 } < \cdots < i _ { m } \right)$, and if $a _ { i _ { 1 } } < a _ { i _ { 2 } } < \cdots < a _ { i _ { m } }$, then the new sequence $a _ { i _ { 1 } } , a _ { i _ { 2 } } , \cdots, a _ { i _ { m } }$ is called an increasing subsequence of length $m$ of $\left\{ a _ { n } \right\}$. By convention, any single term of the sequence $\left\{ a _ { n } \right\}$ is an increasing subsequence of length 1 of $\left\{ a _ { n } \right\}$. (I) Write out an increasing subsequence of length 4

The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { n + 2 } + ( - 1 ) ^ { n } a _ { n } = 3 n - 1$ . The sum of the first 16 terms is 540. Then $a _ { 1 } =$ $\_\_\_\_$.

The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 3 , a _ { n + 1 } = 3 a _ { n } - 4 n$ .
(1) Calculate $a _ { 2 } , a _ { 3 }$ , conjecture the general term formula of $\left\{ a _ { n } \right\}$ and prove it;
(2) Find the sum $S _ { n }$ of the first $n$ terms of the sequence $\left\{ 2 ^ { n } a _ { n } \right\}$ .

In the sequence $\left\{ a_{n} \right\}$ , let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ , $S_{5} = 5S_{3} - 4$ , then $S_{4} =$
A. $7$
B. $9$
C. $15$
D. $20$

In the sequence $\left\{ a_{n} \right\}$ , $a_{2} = 1$ . Let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ . $2S_{n} = na_{n}$ .
(1) Find the general term formula for $\left\{ a_{n} \right\}$ ;
(2) Find the sum $T_{n}$ of the first $n$ terms of the sequence $\left\{ \frac{a_{n} + 1}{2^{n}} \right\}$ .

Given that the domain of function $f ( x )$ is $\mathbb { R }$ , $f ( x ) > f ( x - 1 ) + f ( x - 2 )$ , and when $x < 3$ , $f ( x ) = x$ , then the following conclusion that must be correct is
A. $f ( 10 ) > 100$
B. $f ( 20 ) > 1000$
C. $f ( 10 ) < 1000$
D. $f ( 20 ) < 10000$

Given $M = \left\{ k \mid a _ { k } = b _ { k } \right\}$, where $a _ { n }, b _ { n }$ are not constant sequences and all terms are distinct. Which of the following is correct? \_\_\_
(1) If $a _ { n }, b _ { n }$ are both arithmetic sequences, then $M$ has at most one element;
(2) If $a _ { n }, b _ { n }$ are both geometric sequences, then $M$ has at most three elements;
(3) If $a _ { n }$ is an arithmetic sequence and $b _ { n }$ is a geometric sequence, then $M$ has at most three elements;
(4) If $a _ { n }$ is monotonically increasing and $b _ { n }$ is monotonically decreasing, then $M$ has at most one element.

Given hyperbola $C : x ^ { 2 } - y ^ { 2 } = m ( m > 0 )$, point $P _ { 1 } ( 5,4 )$ is on $C$, and $k$ is a constant with $0 < k < 1$. Through $P_n$ on the right branch of $C$, draw a line with slope $k$; this line intersects $C$ at another point $Q_n$ on the left branch. The reflection of $Q_n$ across the $y$-axis gives $P_{n+1}$ on the right branch.
(1) Find the coordinates of $P_2$ when $k = \frac{1}{2}$.
(2) Prove that $\{x_n - y_n\}$ is a geometric sequence.
(3) Prove that $S_n = \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1})$ is a constant independent of $n$.

Let the sequence $\{a_n\}$ satisfy $a_1 = 3$, $\frac{a_{n+1}}{n} = \frac{a_n}{n+1} + \frac{1}{n(n+1)}$.
(1) Prove that $\{na_n\}$ is an arithmetic sequence.
(2) Let $f(x) = a_1 x + a_2 x^2 + \cdots + a_m x^m$. Find $f'(-2)$.

(15 points) Let the sequence $\{a_n\}$ satisfy $a_1 = 3$, $\frac{a_{n+1}}{n} = \frac{a_n}{n+1} + \frac{1}{n(n+1)}$.
(1) Prove that $\{na_n\}$ is an arithmetic sequence.
(2) Let $f(x) = a_1 x + a_2 x^2 + \cdots + a_m x^m$. Find $f'(-2)$.

We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$, i.e. $f(x) = \phi_0(f(2x))$ for $x\in[0,\frac{1}{2}]$ and $f(x) = \phi_1(f(2x-1))$ for $x\in]\frac{1}{2},1]$.
Let $(r_n)_{n\geq 1}\in\{0,1\}^{\mathbf{N}^*}$.
a) Show that the series with general term $\frac{r_n}{2^n}$ converges and that its sum $x$ belongs to $[0,1]$.
b) By setting for every natural number $p$, $x_p = \sum_{n=1}^{\infty}\frac{r_{n+p}}{2^n}$, prove the relation: $$f(x) = \phi_{r_1}\circ\phi_{r_2}\circ\ldots\phi_{r_p}\left(f\left(x_p\right)\right)$$ for every non-zero natural number $p$.

Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Determine two real numbers $a$ and $b$ such that $$\forall x \in \mathbb{R}, \forall n \in \mathbb{N}^*, T_{n+2}(x) = a x T_{n+1}(x) + b T_n(x)$$

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$.
Show that the numerical series $\sum_{n \in \mathbb{N}} n^j \alpha_n$ is convergent.

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$. We set, for $n \in \mathbb{N}$, $$R_n(j) = \sum_{p=n+1}^{+\infty} p^j \alpha_p$$
Show that the sequence $(R_n(j))_{n \in \mathbb{N}}$ has rapid decay.

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Show that, for all $x \in [-1,1]$, the series $$\sum_{n \geqslant 0} \alpha_n F_n(x)$$ is convergent.

For every real $\alpha > 1$, show that $1 \leqslant S ( \alpha ) \leqslant 1 + \frac { 1 } { \alpha - 1 }$, where $S ( \alpha ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { \alpha } }$.

For every real $\alpha$ strictly greater than 1 and for every non-zero natural integer $n$, we set $R _ { n } ( \alpha ) = \sum _ { k = n } ^ { + \infty } \frac { 1 } { k ^ { \alpha } }$. Using the inequality from question I.A.1, show that $R _ { n } ( \alpha ) = \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } } + O \left( \frac { 1 } { n ^ { \alpha } } \right)$.

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$. By applying to $f$ the Taylor formula with integral remainder at order 2, show that for every $k \in \mathbb { N } ^ { * } , f ( k + 1 ) - f ( k ) = \frac { 1 } { k ^ { \alpha } } - \frac { \alpha } { 2 } \frac { 1 } { k ^ { \alpha + 1 } } + A _ { k }$ where $A _ { k }$ is a real number satisfying $0 \leqslant A _ { k } \leqslant \frac { \alpha ( \alpha + 1 ) } { 2 k ^ { \alpha + 2 } }$.

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$, where $\alpha$ is a real number strictly greater than 1. We fix a non-zero natural integer $p$ and denote by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { 2 p - 1 } f ^ { ( 2 p - 1 ) }$. For every $k \in \mathbb { N } ^ { * }$, we set $R ( k ) = g ( k + 1 ) - g ( k ) - f ^ { \prime } ( k )$ so that $g ( k + 1 ) - g ( k ) = f ^ { \prime } ( k ) + R ( k )$.
By applying to $g$ the Taylor formula with integral remainder at order $2 p$, show that there exists a real number $A$ such that for every $k \in \mathbb { N } ^ { * } , | R ( k ) | \leqslant A k ^ { - ( 2 p + \alpha ) }$.