We are given the sequence $( u _ { n } )$ defined by: $u _ { 0 } = 0$ and for every natural integer $n$, $u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 1$. The sequence $\left( v _ { n } \right)$, defined for every natural integer $n$ by $v _ { n } = u _ { n } - 2$, is: a. arithmetic with common difference $- 2$; b. geometric with common ratio $- 2$; c. arithmetic with common difference $1$; d. geometric with common ratio $\frac { 1 } { 2 }$.

Consider the sequence $(u_n)$ defined for every natural number $n$ by: $$u_n = \mathrm{e}^{2n+1}$$ The sequence $(u_n)$ is: a. arithmetic with common difference 2; b. geometric with common ratio e; c. geometric with common ratio $\mathrm{e}^2$; d. convergent to e.

For questions 3. and 4., consider the sequence $(u_n)$ defined on $\mathbb{N}$ by: $$u_0 = 15 \text{ and for every natural number } n : u_{n+1} = 1{,}2\, u_n + 12.$$
Consider the sequence $(v_n)$ defined on $\mathbb{N}$ by: $v_n = u_n + 60$.
The sequence $(v_n)$ is: a. a decreasing sequence; b. a geometric sequence with common ratio 1,2; c. an arithmetic sequence with common difference 60; d. a sequence that is neither geometric nor arithmetic.

Suppose $a_0, a_1, a_2, a_3, \ldots$ is an arithmetic progression with $a_0$ and $a_1$ positive integers. Let $g_0, g_1, g_2, g_3, \ldots$ be the geometric progression such that $g_0 = a_0$ and $g_1 = a_1$.
Statements
(1) We must have $\left(a_5\right)^2 \geq a_0 a_{10}$.
(2) The sum $a_0 + a_1 + \cdots + a_{10}$ must be a multiple of the integer $a_5$.
(3) If $\sum_{i=0}^{\infty} a_i$ is $+\infty$ then $\sum_{i=0}^{\infty} g_i$ is also $+\infty$.
(4) If $\sum_{i=0}^{\infty} g_i$ is finite then $\sum_{i=0}^{\infty} a_i$ is $-\infty$.

For an infinite geometric sequence $\left\{ a _ { n } \right\}$, choose all correct statements from \textless Remarks\textgreater. [3 points]
\textless Remarks\textgreater ㄱ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges, then $\sum _ { n = 1 } ^ { \infty } a _ { 2 n }$ also converges. ㄴ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges, then $\sum _ { n = 1 } ^ { \infty } a _ { 2 n }$ also diverges. ㄷ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges, then $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } + \frac { 1 } { 2 } \right)$ also converges.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

Let $S_n$ denote the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, and let $q$ be the common ratio of $\{a_n\}$, $q > 0$. If $S_3 = 7$, $a_3 = 1$, then
A. $q = \frac{1}{2}$
B. $a_5 = \frac{1}{9}$
C. $S_5 = 8$
D. $a_n + S_n = 8$

If $I _ { n } = \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \cot ^ { n } x \, d x$, then
(1) $I _ { 2 } + I _ { 4 } , \left( I _ { 3 } + I _ { 5 } \right) ^ { 2 } , I _ { 4 } + I _ { 6 }$ are in G.P.
(2) $I _ { 2 } + I _ { 4 } , I _ { 3 } + I _ { 5 } , I _ { 4 } + I _ { 6 }$ are in A.P.
(3) $\frac { 1 } { I _ { 2 } + I _ { 4 } } , \frac { 1 } { I _ { 3 } + I _ { 5 } } , \frac { 1 } { I _ { 4 } + I _ { 6 } }$ are in A.P.
(4) $\frac { 1 } { I _ { 2 } + I _ { 4 } } , \frac { 1 } { I _ { 3 } + I _ { 5 } } , \frac { 1 } { I _ { 4 } + I _ { 6 } }$ are in G.P.

The parabolas: $ax^{2} + 2bx + cy = 0$ and $dx^{2} + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then
(1) $d, e, f$ are in A.P.
(2) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in G.P.
(3) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.
(4) $d, e, f$ are in G.P.

A sequence $a _ { 1 } , a _ { 2 } , \cdots$ where the odd-indexed terms form a geometric sequence with common ratio $\frac { 1 } { 3 }$ , and the even-indexed terms form a geometric sequence with common ratio $\frac { 1 } { 2 }$ , with $a _ { 1 } = 3 , a _ { 2 } = 2$ . Select the correct options.
(1) $a _ { 4 } > a _ { 5 } > a _ { 6 } > a _ { 7 }$
(2) $\frac { a _ { 10 } } { a _ { 11 } } > 10$
(3) $\lim _ { n \rightarrow \infty } a _ { n } = 0$
(4) $\lim _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n } } = 0$
(5) $\sum _ { n = 1 } ^ { 100 } a _ { n } > 9$

Given that positive real numbers $a , b , c , d , e$ form a geometric sequence with $a < b < c < d < e$, select the options that form a geometric sequence.
(1) $a , - b , c , - d , e$
(2) $e , d , c , b , a$
(3) $\log a , \log b , \log c , \log d , \log e$
(4) $3 ^ { a } , 3 ^ { b } , 3 ^ { c } , 3 ^ { d } , 3 ^ { e }$
(5) $a b c , b c d , c d e$