The purpose of this exercise is to study sequences of positive terms whose first term $u_0$ is strictly greater than 1 and possessing the following property: for every natural number $n > 0$, the sum of the first $n$ consecutive terms equals the product of the first $n$ consecutive terms. We admit that such a sequence exists and we denote it $(u_n)$. It therefore satisfies three properties:

• $u_0 > 1$,
• for all $n \geqslant 0, u_n \geqslant 0$,
• for all $n > 0, u_0 + u_1 + \cdots + u_{n-1} = u_0 \times u_1 \times \cdots \times u_{n-1}$.

1. We choose $u_0 = 3$. Determine $u_1$ and $u_2$.
2. For every integer $n > 0$, we denote $s_n = u_0 + u_1 + \cdots + u_{n-1} = u_0 \times u_1 \times \cdots \times u_{n-1}$.
   In particular, $s_1 = u_0$. a. Verify that for every integer $n > 0, s_{n+1} = s_n + u_n$ and $s_n > 1$. b. Deduce that for every integer $n > 0$, $$u_n = \frac{s_n}{s_n - 1}$$ c. Show that for all $n \geqslant 0, u_n > 1$.
3. Using the algorithm opposite, we want to calculate the term $u_n$ for a given value of $n$. a. Copy and complete the processing part of the algorithm:
   

   Input:	Enter $n$
   	Enter $u$
   Processing:	$s$ takes the value $u$
   	For $i$ going from 1 to $n$:
   	$u$ takes the value $\ldots$
   	$s$ takes the value $\ldots$
   	End For
   Output:	Display $u$

   
   b. The table below gives values rounded to the nearest thousandth of $u_n$ for different values of the integer $n$:
   

   $n$	0	5	10	20	30	40
   $u_n$	3	1.140	1.079	1.043	1.030	1.023

   
   What conjecture can be made about the convergence of the sequence $(u_n)$?
4. a. Justify that for every integer $n > 0, s_n > n$. b. Deduce the limit of the sequence $(s_n)$ then that of the sequence $(u_n)$.

In rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 } , \mathrm {~N} _ { 1 }$ be the midpoints of segments $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ and $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ respectively.
Draw a circular sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ with center $\mathrm { N } _ { 1 }$, radius $\overline { \mathrm { B } _ { 1 } \mathrm {~N} _ { 1 } }$, and central angle $\frac { \pi } { 2 }$, and draw a circular sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ with center $\mathrm { D } _ { 1 }$, radius $\overline { \mathrm { C } _ { 1 } \mathrm { D } _ { 1 } }$, and central angle $\frac { \pi } { 2 }$.
The region enclosed by the arc $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ of sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and the region enclosed by the arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ of sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ are shaded to form a checkmark shape, and the resulting figure is called $R _ { 1 }$.
In figure $R _ { 1 }$, a rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { A } _ { 2 }$ on segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$, and two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ on side $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ such that $\overline { \mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } } : \overline { \mathrm { A } _ { 2 } \mathrm { D } _ { 2 } } = 1 : 2$. A checkmark shape is shaded in rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ using the same method as for figure $R _ { 1 }$, and the resulting figure is called $R _ { 2 }$.
Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 25 } { 19 } \left( \frac { \pi } { 2 } - 1 \right)$
(2) $\frac { 5 } { 4 } \left( \frac { \pi } { 2 } - 1 \right)$
(3) $\frac { 25 } { 21 } \left( \frac { \pi } { 2 } - 1 \right)$
(4) $\frac { 25 } { 22 } \left( \frac { \pi } { 2 } - 1 \right)$
(5) $\frac { 25 } { 23 } \left( \frac { \pi } { 2 } - 1 \right)$

For a natural number $k$, $$a _ { k } = \lim _ { n \rightarrow \infty } \frac { \left( \frac { 6 } { k } \right) ^ { n + 1 } } { \left( \frac { 6 } { k } \right) ^ { n } + 1 }$$ Find the value of $\sum _ { k = 1 } ^ { 10 } k a _ { k }$. [4 points]

As shown in the figure, for a square ABCD with side length 5, let the five division points of diagonal BD be $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 }$ in order from point B. Draw squares with diagonals $\mathrm { BP } _ { 1 } , \mathrm { P } _ { 2 } \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 } \mathrm { D }$ and circles with diameters $\mathrm { P } _ { 1 } \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } \mathrm { P } _ { 4 }$, then color the figure-eight-shaped region to obtain figure $R _ { 1 }$. In figure $R _ { 1 }$, let $\mathrm { Q } _ { 1 }$ be the vertex of the square with diagonal $\mathrm { P } _ { 2 } \mathrm { P } _ { 3 }$ closest to point A, and $\mathrm { Q } _ { 2 }$ be the vertex closest to point C. Draw squares with diagonals $\mathrm { AQ } _ { 1 }$ and $\mathrm { CQ } _ { 2 }$, and in these 2 new squares, draw figure-eight-shaped figures using the same method as for $R _ { 1 }$ and color them to obtain figure $R _ { 2 }$. In figure $R _ { 2 }$, in the squares with diagonals $\mathrm { AQ } _ { 1 }$ and $\mathrm { CQ } _ { 2 }$, draw figure-eight-shaped figures using the same method as obtaining $R _ { 2 }$ from $R _ { 1 }$ and color them to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [3 points]
(1) $\frac { 24 } { 17 } ( \pi + 3 )$
(2) $\frac { 25 } { 17 } ( \pi + 3 )$
(3) $\frac { 26 } { 17 } ( \pi + 3 )$
(4) $\frac { 24 } { 17 } ( 2 \pi + 1 )$
(5) $\frac { 25 } { 17 } ( 2 \pi + 1 )$

As shown in the figure, there is a circle $O$ with diameter AB of length 4. Let C be the center of the circle, and let D and P be the midpoints of segments AC and BC, respectively. Let E and Q be the points where the perpendicular bisector of segment AC and the perpendicular bisector of segment BC meet the upper semicircle of circle $O$, respectively. Draw a square DEFG with side DE that meets circle $O$ at point A and has diagonal DF, and draw a square PQRS with side PQ that meets circle $O$ at point B and has diagonal PR. Color the $\square$-shaped figure that is the common part of the interior of circle $O$ and the interior of square DEFG, and the $\square$-shaped figure that is the common part of the interior of circle $O$ and the interior of square PQRS to obtain figure $R _ { 1 }$. In figure $R _ { 1 }$, draw circle $O _ { 1 }$ centered at point F with radius $\frac { 1 } { 2 } \overline { \mathrm { DE } }$, and circle $O _ { 2 }$ centered at point R with radius $\frac { 1 } { 2 } \overline { \mathrm { PQ } }$. Color 2 $\square$-shaped figures and 2 $\square$-shaped figures created in the same way as obtaining figure $R _ { 1 }$ on the two circles $O _ { 1 }$ and $O _ { 2 }$ to obtain figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in figure $R _ { n }$ obtained the $n$-th time. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 12 \pi - 9 \sqrt { 3 } } { 10 }$
(2) $\frac { 8 \pi - 6 \sqrt { 3 } } { 5 }$
(3) $\frac { 32 \pi - 24 \sqrt { 3 } } { 15 }$
(4) $\frac { 28 \pi - 21 \sqrt { 3 } } { 10 }$
(5) $\frac { 16 \pi - 12 \sqrt { 3 } } { 5 }$

The sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions for all natural numbers $n$. (가) $a _ { 2 n } = a _ { n } - 1$ (나) $a _ { 2 n + 1 } = 2 a _ { n } + 1$ When $a _ { 20 } = 1$, what is the value of $\sum _ { n = 1 } ^ { 63 } a _ { n }$? [4 points]
(1) 704
(2) 712
(3) 720
(4) 728
(5) 736

For a sequence $\left\{ a _ { n } \right\}$ with first term 1, satisfying for all natural numbers $n$: $$a _ { n + 1 } = \begin{cases} 2 a _ { n } & \left( a _ { n } < 7 \right) \\ a _ { n } - 7 & \left( a _ { n } \geq 7 \right) \end{cases}$$ what is the value of $\sum _ { k = 1 } ^ { 8 } a _ { k }$? [3 points]
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

17.
(1)
By mathematical induction, we can deduce that
$$a _ { n } = \begin{cases} \frac { 3 n - 1 } { 2 } & \text{if } 2 \nmid n \\ \frac { 2 n - 2 } { 2 } & \text{if } 2 \mid n \end{cases}$$
Thus $b _ { n } = a _ { 2 n } = 3 n - 1$ for $n \in \mathbb { Z } ^ { + }$, with $b _ { 1 } = 2, b _ { 2 } = 5$.
(2)
$$\begin{gathered} \sum _ { k = 1 } ^ { 20 } a _ { k } = \sum _ { k = 1 } ^ { 10 } a _ { 2 k - 1 } + \sum _ { k = 1 } ^ { 10 } a _ { 2 k } \\ = \sum _ { k = 1 } ^ { 10 } ( 3 k - 2 ) + \sum _ { k = 1 } ^ { 10 } ( 3 k - 1 ) \\ = 6 \sum _ { k = 1 } ^ { 10 } k - 30 = 300 \end{gathered}$$

Using the results of II.C.6, recover the relation $$V_n(z) = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n-j}{j} (2z)^{n-2j} (-1)^j$$

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. For all $x \in \mathcal{D}_{f}$, verify that $x + k \in \mathcal{D}_{f}$, then calculate $f(x+k) - f(x)$.

Let $f(a, b)$ be a function satisfying $f(a, b) = f(a, c) + f(c, b) - 2f(a,c)f(c,b)$ with $f(99, 100) = 1/3$. Find $f(1, 100)$.

The sum $\sum _ { r = 1 } ^ { 10 } \left( r ^ { 2 } + 1 \right) \times ( r ! )$, is equal to:
(1) $11 \times ( 11 ! )$
(2) $10 \times ( 11 ! )$
(3) $(11)!$
(4) $101 \times ( 10 ! )$

Consider an arithmetic series and a geometric series having four initial terms from the set $\{ 11,8,21,16,26,32,4 \}$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to $\_\_\_\_$.

Let $\left\{ a _ { n } \right\} _ { n = 1 } ^ { \infty }$ be a sequence such that $a _ { 1 } = 1 , a _ { 2 } = 1$ and $a _ { n + 2 } = 2 a _ { n + 1 } + a _ { n }$ for all $n \geq 1$. Then the value of $47 \sum _ { n = 1 } ^ { \infty } \left( \frac { a _ { n } } { 2 ^ { 3 n } } \right)$ is equal to $\underline{\hspace{1cm}}$.

Let $\left\{ a _ { n } \right\} _ { n = 0 } ^ { \infty }$ be a sequence such that $a _ { 0 } = a _ { 1 } = 0$ and $a _ { n + 2 } = 2 a _ { n + 1 } - a _ { n } + 1$ for all $n \geq 0$. Then, $\sum _ { n = 2 } ^ { \infty } \frac { a _ { n } } { 7 ^ { n } }$ is equal to
(1) $\frac { 6 } { 343 }$
(2) $\frac { 7 } { 216 }$
(3) $\frac { 8 } { 343 }$
(4) $\frac { 49 } { 216 }$

Consider the sequence $a_1, a_2, a_3, \ldots$ such that $a_1 = 1$, $a_2 = 2$ and $a_{n+2} = \frac{2}{a_{n+1}} + a_n$ for $n = 1, 2, 3, \ldots$ If $\frac{a_1 + \frac{1}{a_2}}{a_3} \cdot \frac{a_2 + \frac{1}{a_3}}{a_4} \cdot \frac{a_3 + \frac{1}{a_4}}{a_5} \cdots \frac{a_{30} + \frac{1}{a_{31}}}{a_{32}} = 2^\alpha \binom{61}{31}$ then $\alpha$ is equal to
(1) $-30$
(2) $-31$
(3) $-60$
(4) $-61$

Let $a _ { 1 } = b _ { 1 } = 1$ and $a _ { n } = a _ { n - 1 } + ( n - 1 ) , b _ { n } = b _ { n - 1 } + \mathrm { a } _ { n - 1 } , \forall n \geq 2$. If $\mathrm { S } = \sum _ { \mathrm { n } = 1 } ^ { 10 } \left( \frac { b _ { n } } { 2 ^ { n } } \right)$ and $\mathrm { T } = \sum _ { n = 1 } ^ { 8 } \frac { \mathrm { n } } { 2 ^ { n - 1 } }$ then $2 ^ { 7 } ( 2 S - T )$ is equal to $\_\_\_\_$

Let $\left\langle a _ { n } \right\rangle$ be a sequence such that $a _ { 1 } + a _ { 2 } + \ldots + a _ { n } = \frac { n ^ { 2 } + 3 n } { ( n + 1 ) ( n + 2 ) }$. If $28 \sum _ { k = 1 } ^ { 10 } \frac { 1 } { a _ { k } } = p _ { 1 } p _ { 2 } p _ { 3 } \ldots p _ { m }$, where $p _ { 1 } , p _ { 2 } , \ldots p _ { m }$ are the first $m$ prime numbers, then $m$ is equal to
(1) 5
(2) 8
(3) 6
(4) 7

Let $\left\langle a _ { \mathrm { n } } \right\rangle$ be a sequence such that $a _ { 0 } = 0 , a _ { 1 } = \frac { 1 } { 2 }$ and $2 a _ { \mathrm { n } + 2 } = 5 a _ { \mathrm { n } + 1 } - 3 a _ { \mathrm { n } } , \mathrm { n } = 0,1,2,3 , \ldots$ Then $\sum _ { \mathrm { k } = 1 } ^ { 100 } a _ { k }$ is equal to
(1) $3 a _ { 99 } - 100$
(2) $3 \mathrm { a } _ { 100 } - 100$
(3) $3 a _ { 99 } + 100$
(4) $3 \mathrm { a } _ { 100 } + 100$

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be a geometric sequence with first term 3 and common ratio $3\sqrt{3}$. Select the number of terms $n$ that satisfy the inequality
$$\log_{3} a_{1} - \log_{3} a_{2} + \log_{3} a_{3} - \log_{3} a_{4} + \ldots + (-1)^{n+1} \log_{3} a_{n} > 18$$
among the possible options.
(1) 23
(2) 24
(3) 25
(4) 26
(5) 27

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