Four friends, each with 100 coins, created a game, in which each one assumes one of four positions, $1, 2, 3$, or $4$, indicated in the figure, and remains there until the end.
The development of the game takes place in rounds and, in all of them, each player transfers and receives a quantity of coins, as follows:

• the player in position 1 transfers 1 coin to the player in position 2;
• the player in position 2 transfers 2 coins to the player in position 3;
• the player in position 3 transfers 3 coins to the player in position 4;
• the player in position 4 transfers 4 coins to the player in position 1, completing the round.

At the end of round $n$, what is the algebraic expression that represents the number of coins of the player in position 1?
(A) $103 + 4n$
(B) $103 + 3n$
(C) $100 + 4n$
(D) $100 + 3n$
(E) $99 + 4n$

(12 points)
Let $\{a_n\}$ be a sequence with $a_1 + a_2 = 2$.
(1) If $\{a_n\}$ is an arithmetic sequence and $a_1 + a_3 = 5$, find the general formula for $\{a_n\}$.
(2) If $\{a_n\}$ is a geometric sequence and $T_n$ denotes the sum of the first $n$ terms of another related sequence with $T_n = 21$, find $S_n$.

9. Let $S _ { n }$ denote the sum of the first $n$ terms of an arithmetic sequence $\left\{ a _ { n } \right\}$ . Given $S _ { 4 } = 0 , a _ { 5 } = 5$ , then
A. $a _ { n } = 2 n - 5$
B. $a _ { n } = 3 n - 10$
C. $S _ { n } = 2 n ^ { 2 } - 8 n$
D. $S _ { n } = \frac { 1 } { 2 } n ^ { 2 } - 2 n$

Let the sum of the first $n$ terms of a non-constant A.P., $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { n }$ be $50 n + \frac { n ( n - 7 ) } { 2 } A$, where $A$ is a constant. If $d$ is the common difference of this A.P., then the ordered pair $\left( d , a _ { 50 } \right)$ is equal to
(1) $( 50,50 + 46 A )$
(2) $( A , 50 + 45 A )$
(3) $( 50,50 + 45 A )$
(4) $( A , 50 + 46 A )$