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

As shown in the figure, there is a rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { AB } _ { 1 } } = 2$ and $\overline { \mathrm { AD } _ { 1 } } = 4$. Let $\mathrm { E } _ { 1 }$ be the point that divides segment $\mathrm { AD } _ { 1 }$ internally in the ratio $3 : 1$, and let $\mathrm { F } _ { 1 }$ be a point inside rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ such that $\overline { \mathrm { F } _ { 1 } \mathrm { E } _ { 1 } } = \overline { \mathrm { F } _ { 1 } \mathrm { C } _ { 1 } }$ and $\angle \mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } = \frac { \pi } { 2 }$. Triangle $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 }$ is drawn. The figure obtained by shading quadrilateral $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ is called $R _ { 1 }$. In figure $R _ { 1 }$, a rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { B } _ { 2 }$ on segment $\mathrm { AB } _ { 1 }$, point $\mathrm { C } _ { 2 }$ on segment $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on segment $\mathrm { AE } _ { 1 }$, and point A, such that $\overline { \mathrm { AB } _ { 2 } } : \overline { \mathrm { AD } _ { 2 } } = 1 : 2$. Using the same method as for obtaining figure $R _ { 1 }$, triangle $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 }$ is drawn in rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and quadrilateral $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is shaded to obtain figure $R _ { 2 }$. 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) $\frac { 441 } { 103 }$
(2) $\frac { 441 } { 109 }$
(3) $\frac { 441 } { 115 }$
(4) $\frac { 441 } { 121 }$
(5) $\frac { 441 } { 127 }$

For a constant $k$ with $k > 1$, there is a sequence $\left\{ a _ { n } \right\}$ satisfying the following conditions.
For all natural numbers $n$, $a _ { n } < a _ { n + 1 }$ and the slope of the line passing through two points $\mathrm { P } _ { n } \left( a _ { n } , 2 ^ { a _ { n } } \right)$ and $\mathrm { P } _ { n + 1 } \left( a _ { n + 1 } , 2 ^ { a _ { n + 1 } } \right)$ on the curve $y = 2 ^ { x }$ is $k \times 2 ^ { a _ { n } }$.
Let $\mathrm { Q } _ { n }$ be the point where the line passing through $\mathrm { P } _ { n }$ parallel to the $x$-axis and the line passing through $\mathrm { P } _ { n + 1 }$ parallel to the $y$-axis meet, and let $A _ { n }$ be the area of triangle $\mathrm { P } _ { n } \mathrm { Q } _ { n } \mathrm { P } _ { n + 1 }$. The following is the process of finding $A _ { n }$ when $a _ { 1 } = 1$ and $\frac { A _ { 3 } } { A _ { 1 } } = 16$.
Since the slope of the line passing through two points $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ is $k \times 2 ^ { a _ { n } }$, $$2 ^ { a _ { n + 1 } - a _ { n } } = k \left( a _ { n + 1 } - a _ { n } \right) + 1$$ Thus, for all natural numbers $n$, $a _ { n + 1 } - a _ { n }$ is a solution of the equation $2 ^ { x } = k x + 1$. Since $k > 1$, the equation $2 ^ { x } = k x + 1$ has exactly one positive real root $d$. Therefore, for all natural numbers $n$, $a _ { n + 1 } - a _ { n } = d$, and the sequence $\left\{ a _ { n } \right\}$ is an arithmetic sequence with common difference $d$. Since the coordinates of point $\mathrm { Q } _ { n }$ are $\left( a _ { n + 1 } , 2 ^ { a _ { n } } \right)$, $$A _ { n } = \frac { 1 } { 2 } \left( a _ { n + 1 } - a _ { n } \right) \left( 2 ^ { a _ { n + 1 } } - 2 ^ { a _ { n } } \right)$$ Since $\frac { A _ { 3 } } { A _ { 1 } } = 16$, the value of $d$ is (가), and the general term of the sequence $\left\{ a _ { n } \right\}$ is $$a _ { n } = \text { (나) }$$ Therefore, for all natural numbers $n$, $A _ { n } =$ (다).
When the number corresponding to (가) is $p$, and the expressions corresponding to (나) and (다) are $f ( n )$ and $g ( n )$ respectively, what is the value of $p + \frac { g ( 4 ) } { f ( 2 ) }$? [4 points]
(1) 118
(2) 121
(3) 124
(4) 127
(5) 130

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) $\frac { 32 } { 7 }$

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. $\frac { 1 } { 8 }$

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 $\left\{ b _ { n } \right\}$.

The answer to a problem is: The sequence is $1, 1, 2, 2^0, 2^1$, and the next three terms are $2^0, 2^1, 2^2$, and so on. The first term is $2^0$. Find the smallest positive integer $N$ such that $N > 100$ and the sum of the first $N$ terms of this sequence is an integer power of 2.
A. 440
B. 330
C. 220
D. 110

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

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

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 $S _ { 5 } =$ $\_\_\_\_$ .

In the sequence $\left\{ a _ { n } \right\}$ , $a _ { 1 } = 2 , a _ { m + n } = a _ { m } a _ { n }$ . If $a _ { k + 1 } + a _ { k + 2 } + \cdots + a _ { k + 10 } = 2 ^ { 15 } - 2 ^ { 5 }$ , then $k =$
A． 2
B． 3
C． 4
D． 5

Let $\left\{ a _ { n } \right\}$ be a geometric sequence with $a _ { 1 } + a _ { 2 } + a _ { 3 } = 1 , a _ { 2 } + a _ { 3 } + a _ { 4 } = 2$ , then $a _ { 6 } + a _ { 7 } + a _ { 8 } =$
A. 12
B. 24
C. 30
D. 32

Let the geometric sequence $\left\{ a _ { n } \right\}$ satisfy $a _ { 1 } + a _ { 2 } = 4 , a _ { 3 } - a _ { 1 } = 8$ .
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ \log _ { 3 } a _ { n } \right\}$. If $S _ { m } + S _ { m + 1 } = S _ { m + 3 }$, find $m$ .

7. For a geometric sequence $\{a_n\}$ with common ratio $q$ and sum of the first $n$ terms $S_n$, let Proposition A: $q > 0$. Proposition B: $\{S_n\}$ is an increasing sequence. Then
A. A is a sufficient but not necessary condition for B
B. A is a necessary but not sufficient condition for B
C. A is a necessary and sufficient condition for B
D. A is neither a sufficient nor a necessary condition for B