For a geometric sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 3 , a _ { 2 } = 1$, what is the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } \right) ^ { 2 }$? [3 points]
(1) $\frac { 81 } { 8 }$
(2) $\frac { 83 } { 8 }$
(3) $\frac { 85 } { 8 }$
(4) $\frac { 87 } { 8 }$
(5) $\frac { 89 } { 8 }$

For a geometric sequence $\left\{ a _ { n } \right\}$ with a non-zero first term, $$a _ { 3 } = 4 a _ { 1 } , \quad a _ { 7 } = \left( a _ { 6 } \right) ^ { 2 }$$ what is the value of the first term $a _ { 1 }$? [3 points]
(1) $\frac { 1 } { 16 }$
(2) $\frac { 1 } { 8 }$
(3) $\frac { 3 } { 16 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 5 } { 16 }$

For a geometric sequence $\left\{ a _ { n } \right\}$ with first term 1 and common ratio $r$ ($r > 1$), let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. When $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { S _ { n } } = \frac { 3 } { 4 }$, find the value of $r$. [3 points]

When three numbers $\frac { 9 } { 4 } , a , 4$ form a geometric sequence in this order, what is the value of the positive number $a$? [3 points]
(1) $\frac { 8 } { 3 }$
(2) 3
(3) $\frac { 10 } { 3 }$
(4) $\frac { 11 } { 3 }$
(5) 4

For a geometric sequence $\left\{ a _ { n } \right\}$ with first term 7, let $S _ { n }$ denote the sum of the first $n$ terms. $$\frac { S _ { 9 } - S _ { 5 } } { S _ { 6 } - S _ { 2 } } = 3$$ Find the value of $a _ { 7 }$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms, $$\frac { a _ { 16 } } { a _ { 14 } } + \frac { a _ { 8 } } { a _ { 7 } } = 12$$ Find the value of $\frac { a _ { 3 } } { a _ { 1 } } + \frac { a _ { 6 } } { a _ { 3 } }$. [3 points]

For a geometric sequence $\left\{ a _ { n } \right\}$ with first term $\frac { 1 } { 8 }$, if $\frac { a _ { 3 } } { a _ { 2 } } = 2$, what is the value of $a _ { 5 }$? [2 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 4

A sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions.
(a) $\left| a _ { 1 } \right| = 2$
(b) For all natural numbers $n$, $\left| a _ { n + 1 } \right| = 2 \left| a _ { n } \right|$.
(c) $\sum _ { n = 1 } ^ { 10 } a _ { n } = - 14$ Find the value of $a _ { 1 } + a _ { 3 } + a _ { 5 } + a _ { 7 } + a _ { 9 }$. [4 points]

A geometric sequence $\left\{ a _ { n } \right\}$ with positive common ratio satisfies $$a _ { 2 } + a _ { 4 } = 30 , \quad a _ { 4 } + a _ { 6 } = \frac { 15 } { 2 }$$ What is the value of $a _ { 1 }$? [3 points]
(1) 48
(2) 56
(3) 64
(4) 72
(5) 80

Let $S_n$ denote the sum of the first $n$ terms of a geometric sequence $\{a_n\}$. $$S_4 - S_2 = 3a_4, \quad a_5 = \frac{3}{4}$$ Find the value of $a_1 + a_2$. [3 points]
(1) 27
(2) 24
(3) 21
(4) 18
(5) 15

A geometric sequence $\left\{ a_{n} \right\}$ with first term and common ratio both equal to a positive number $k$ satisfies $$\frac{a_{4}}{a_{2}} + \frac{a_{2}}{a_{1}} = 30$$ What is the value of $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

A geometric sequence $\left\{ a_{n} \right\}$ satisfies $$\sum_{n=1}^{\infty} \left(\left| a_{n} \right| + a_{n}\right) = \frac{40}{3}, \quad \sum_{n=1}^{\infty} \left(\left| a_{n} \right| - a_{n}\right) = \frac{20}{3}$$ The inequality $$\lim_{n \rightarrow \infty} \sum_{k=1}^{2n} \left((-1)^{\frac{k(k+1)}{2}} \times a_{m+k}\right) > \frac{1}{700}$$ is satisfied. What is the sum of all natural numbers $m$ satisfying this inequality? [4 points]

A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $$2 \left( a _ { 1 } + a _ { 4 } + a _ { 7 } \right) = a _ { 4 } + a _ { 7 } + a _ { 10 } = 6$$ What is the value of $a _ { 10 }$? [4 points]
(1) $\frac { 22 } { 7 }$
(2) $\frac { 24 } { 7 }$
(3) $\frac { 26 } { 7 }$
(4) $\frac { 30 } { 7 }$
(5) $\frac { 32 } { 7 }$

12. A set of quantities that can uniquely determine a sequence is called the ``fundamental quantities'' of that sequence. For an infinite geometric sequence $\left\{ a _ { n } \right\}$ with common ratio $q$, among the following four groups of quantities, which group(s) can definitely serve as the ``fundamental quantities'' of the sequence? (Write all group numbers that satisfy the requirement)
(1) $S _ { 1 }$ and $S _ { 2 }$;
(2) $a _ { 2 }$ and $S _ { 3 }$;
(3) $a _ { 1 }$ and $a _ { n }$;
(4) $q$ and $a _ { n }$.
Here $n$ is an integer greater than 1, and $S _ { n }$ is the sum of the first $n$ terms of $\left\{ a _ { n } \right\}$. II. Multiple-Choice Questions (Total Score: 16 points, 4 points each)

A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 3 , a _ { 1 } + a _ { 3 } + a _ { 5 } = 21$ , then $a _ { 3 } + a _ { 5 } + a _ { 7 } =$
(A) $21$
(B) $42$
(C) $63$
(D) $84$

9. Given a geometric sequence $\left\{ a _ { n } \right\}$ satisfying $a _ { 1 } = \frac { 1 } { 4 } , a _ { 3 } a _ { 5 } = 4 \left( a _ { 4 } - 1 \right)$, then $a _ { 2 } =$
A. $2$
B. $1$
C. $\frac { 1 } { 2 }$
D. $\frac { 1 } { 8 }$

13. Let $s _ { n }$ be the sum of the first n terms of the geometric sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } = 1$ and $3 s _ { 1 } , 2 s _ { 2 } , s _ { 3 }$ form an arithmetic sequence, then $a _ { n } = $ $\_\_\_\_$.

18. The sequence $\left\{ a _ { n } \right\}$ is an increasing geometric sequence with $a _ { 1 } + a _ { 4 } = 9 , a _ { 2 } a _ { 3 } = 8$.
(1) Find the general term formula of the sequence $\left\{ a _ { n } \right\}$;
(2) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$, and $b _ { n } = \frac { a _ { n + 1 } } { S _ { n } S _ { n + 1 } }$. Find the sum $T _ { n }$ of the first $n$ terms of the sequence $\left\{ b _ { n } \right\}$.

In a geometric sequence $\{ a _ { n } \}$, $a _ { 1 } = 1$ and $a _ { 4 } = 4 a _ { 2 }$.
(1) Find the general term formula for $\{ a_n \}$;
(2) Let $S _ { n }$ be the sum of the first $n$ terms of $\{ a_n \}$. If $S _ { m } = 63$, find $m$.

A geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms has the sum of its first 4 terms equal to 15, and $a _ { 5 } = 3 a _ { 3 } + 4 a _ { 1 }$ . Then $a _ { 3 } =$
A. 16
B. 8
C. 4
D. 2

5. A geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms has the sum of its first 4 terms equal to 15, and $a _ { 5 } = 3 a _ { 3 } + 4 a _ { 1 }$ . Then $a _ { 3 } =$
A. 16
B. 8
C. 4
D. 2

6. A geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms has the sum of its first 4 terms equal to 15, and $a _ { 5 } = 3 a _ { 3 } + 4 a _ { 1 }$ . Then $a _ { 3 } =$
A. 16
B. 8
C. 4
D. 2

9. Executing the flowchart below, if the input $\varepsilon$ is 0.01 , then the output value of $S$ equals [Figure]
A. $2 - \frac { 1 } { 2 ^ { 4 } }$
B. $2 - \frac { 1 } { 2 ^ { 5 } }$
C. $2 - \frac { 1 } { 2 ^ { 6 } }$
D. $2 - \frac { 1 } { 2 ^ { 7 } }$

14. Let $S _ { n }$ denote the sum of the first $n$ terms of a geometric sequence $\left\{ a _ { n } \right\}$. If $a _ { 1 } = \frac { 1 } { 3 } , a _ { 4 } ^ { 2 } = a _ { 6 }$, then $S _ { 3 } = \_\_\_\_$.

14. Let $S _ { n }$ denote the sum of the first $n$ terms of a geometric sequence $\left\{ a _ { n } \right\}$ . If $a _ { 1 } = \frac { 1 } { 3 } , a _ { 4 } ^ { 2 } = a _ { 6 }$ , then $S _ { 5 } =$ $\_\_\_\_$ .