Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given that $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$.
(1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence;
(2) If $a _ { 4 }$, $a _ { 7 }$, $a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$.

17. (10 points) Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given that $a _ { 1 } = 1$ and $\left\{ \frac { S _ { n } } { a _ { n } } \right\}$ is an arithmetic sequence with common difference $\frac { 1 } { 3 }$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Prove that $\frac { 1 } { a _ { 1 } } + \frac { 1 } { a _ { 2 } } + \cdots + \frac { 1 } { a _ { n } } < 2$ .

Let $S _ { n }$ denote the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $\frac { 2 S _ { n } } { n } + n = 2 a _ { n } + 1$ .
(1) Prove that $\left\{ a _ { n } \right\}$ is an arithmetic sequence;
(2) If $a _ { 4 } , a _ { 7 } , a _ { 9 }$ form a geometric sequence, find the minimum value of $S _ { n }$ .

Given that the arithmetic sequence $\left\{ a _ { n } \right\}$ has common difference $\frac { 2 \pi } { 3 }$, and the set $S = \left\{ \cos a _ { n } \mid n \in \mathbb{N} ^ { * } \right\}$. If $S = \{ a , b \}$, then $a b =$
A. $- 1$
B. $- \frac { 1 } { 2 }$
C. 0
D. $\frac { 1 } { 2 }$

Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$. If $a _ { 3 } + a _ { 4 } = 7$ and $3 a _ { 2 } + a _ { 5 } = 5$, then $S _ { 10 } =$ $\_\_\_\_$ .

(17 points) Let $m$ be a positive integer. The sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an arithmetic sequence with nonzero common difference. If after removing two terms $a _ { i }$ and $a _ { j } ( i < j )$ , the remaining $4 m$ terms can be evenly divided into $m$ groups, and the 4 numbers in each group form an arithmetic sequence, then the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is called an $( i , j )$ -divisible sequence.
(1) Write out all pairs $( i , j )$ with $1 \leqslant i < j \leqslant 6$ such that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 6 }$ is an $( i , j )$ -divisible sequence;
(2) When $m \geqslant 3$ , prove that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is a $( 2,13 )$ -divisible sequence;
(3) From $1,2 , \cdots , 4 m + 2$ , randomly select two numbers $i$ and $j$ ( $i < j$ ) at once. Let $P _ { m }$ denote the probability that the sequence $a _ { 1 } , a _ { 2 } , \cdots , a _ { 4 m + 2 }$ is an $( i , j )$ -divisible sequence. Prove that $P _ { m } > \frac { 1 } { 8 }$ .

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_3 = 6$, $S_5 = -5$, then $S_6 = $ ( )
A. $-20$
B. $-15$
C. $-10$
D. $-5$

For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ Deduce that $$m_{k} = \begin{cases} C_{k/2} & \text{if } k \text{ is even} \\ 0 & \text{if } k \text{ is odd.} \end{cases}$$

Show that when $n$ tends to $+\infty$, we have an equivalent of the form: $$\sum_{k=1}^{n} \frac{1}{\sqrt{k}} \underset{n \to +\infty}{\sim} \lambda \sqrt{n},$$ where the constant $\lambda$ is to be determined.

Justify that, for all $( p , q ) \in \left( \mathrm { N } ^ { * } \right) ^ { 2 }$, the series $\sum u _ { k }$ converges, where $u_k = \dfrac{(-1)^k}{pk+q}$.

In this question, we set $p = q = 1$. Show that $$\phi _ { 1,1 } ( n ) = \int _ { 0 } ^ { 1 } \frac { 1 } { 1 + t } d t - \int _ { 0 } ^ { 1 } \frac { ( - t ) ^ { n + 1 } } { 1 + t } d t$$ where $\phi_{1,1}(n) = \sum_{k=0}^{n} \dfrac{(-1)^k}{k+1}$.

Deduce the value of $S _ { 1,1 }$, the sum of the congruent-harmonic series with parameters $p=q=1$.

We fix $( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 }$ and set $\alpha _ { p , q } := \dfrac { p } { q }$. We define, for all $t \in \mathbf { R } _ { + }$, the application $I _ { p , q } : \mathbf { R } _ { + } \rightarrow \mathbf { R }$ by $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x$$ For all $x \in [ 0,1 ]$, calculate $\sum _ { k = 0 } ^ { n } \left( - x ^ { \alpha _ { p , q } } \right) ^ { k }$ then deduce that $$\phi _ { p , q } ( n ) = \frac { 1 } { q } \left( \int _ { 0 } ^ { 1 } \frac { 1 } { 1 + x ^ { \alpha _ { p , q } } } d x + ( - 1 ) ^ { n } I _ { p , q } ( n ) \right)$$

Let $r$ and $s$ be two strictly positive natural integers such that $r > s$, and
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = 0 } ^ { + \infty } \left( \frac { 1 } { s + k + 1 } - \frac { 1 } { r + k + 1 } \right)$$
Deduce that
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = s + 1 } ^ { r } \frac { 1 } { k }$$

105. What is the sum of all two-digit natural numbers that are multiples of 7?
(1) $721$ (2) $728$ (3) $735$ (4) $742$

Find the sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4,5$, each digit appearing at most once.

Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then,
(A) $a _ { n } < b _ { n }$ for all $n > 1$
(B) $a _ { n } > b _ { n }$ for all $n > 1$
(C) $a _ { n } = b _ { n }$ for infinitely many $n$
(D) None of the above

The value of $$1 + \frac { 1 } { 1 + 2 } + \frac { 1 } { 1 + 2 + 3 } + \cdots + \frac { 1 } { 1 + 2 + 3 + \cdots 2021 }$$ is
(A) $\frac { 2021 } { 1010 }$.
(B) $\frac { 2021 } { 1011 }$.
(C) $\frac { 2021 } { 1012 }$.
(D) $\frac { 2021 } { 1013 }$.

Let $$S = \frac{1}{\sqrt{10000}} + \frac{1}{\sqrt{10001}} + \cdots + \frac{1}{\sqrt{160000}}$$ Then the largest positive integer not exceeding $S$ is
(A) 200
(B) 400
(C) 600
(D) 800

The limit $$\lim_{n \rightarrow \infty} \frac{2\log 2 + 3\log 3 + \cdots + n\log n}{n^2 \log n}$$ is equal to
(A) 0
(B) $1/4$
(C) $1/2$
(D) 1

2. Let Tr be the rth term of an A.P., for $\mathrm { r } = 1,2,3$, ….. If for some positive integers $\mathrm { m } , \mathrm { n }$ we have $\mathrm { Tm } = 1 / \mathrm { n }$ and $\mathrm { Tn } = 1 / \mathrm { m }$, then Tmn equals:
(A) $1 / \mathrm { mn }$
(B) $1 / m + 1 / n$
(C) 1
(D) 0

9. Let $\mathrm { a } 1 , \mathrm { a } 2 , \ldots \ldots , \mathrm { a } 10$ be in A.P. and h1, h2, ……, h10 be in H.P. If a1 $= \mathrm { h } 1 = 2$ and a10 = $\mathrm { h } 10 = 3$, then a4 h7 is :
(A) 2
(B) 3
(C) 5
(D) 6

22. If the sum of the first $2 n$ terms of the A.P. $2,5,8 , \ldots \ldots \ldots \ldots \ldots$, is equal to the sum of the first $n$ terms of the A.P. $57,59,61 , \ldots .$. ,then $n$ equals:
(A) 10
(B) 12
(C) 11
(D) 13

24. Let the positive numbers $\mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d }$ be in A.P. Then $\mathrm { abc } , \mathrm { abd } , \mathrm { acd } , \mathrm { bcd }$ are:

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(A) NOT in A.P./G.P/H.P.
(B) In A.P
(C) In G.P
(D) In H.P

3. If $\mathrm { a } _ { 1 } \mathrm { a } _ { 2 } , \ldots , \mathrm { a } _ { \mathrm { n } }$ are positive real numbers whose product is a fixed number c , then the minimum value of $a _ { 1 } + a _ { 2 } + \ldots + a _ { n - 1 } + 2 a _ { n }$ is
(A) $\quad n ( 2 c ) ^ { 1 / n }$
(B) $\quad ( n + 1 ) c ^ { 1 / n }$
(C) $\quad 2 \mathrm { nc } ^ { 1 / \mathrm { n } }$
(D) $\quad ( n + 1 ) ( 2 c ) ^ { 1 / n }$