113- The sequence $\left\{\left[\dfrac{(-1)^n}{n}\right]\right\}$, $n = 1, 2, 3, \ldots$ is how?
(1) Converges to $-1$ (2) Converges to zero (3) Divergent -- bounded (4) Divergent

114- The sequence $\left\{\left(1+\dfrac{1}{n^2}\right)^n\right\}$ converges to which number?
(1) $\sqrt{e}$ (2) $\dfrac{1}{2}e$ (3) $1$ (4) $\dfrac{1}{e}$

Let $x_n = \dfrac{1}{(2a)^n}$ for $a > 0$. Which of the following is true?
(A) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$
(B) $x_n \to 0$ if $0 < a < 1/2$ and $x_n \to \infty$ if $a > 1/2$
(C) $x_n \to 1$ for all $a > 0$
(D) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$

For any integer $n \geq 1$, define $a_n = \frac{1000^n}{n!}$. Then the sequence $\{ a_n \}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(a) does not have a maximum
(b) attains maximum at exactly one value of $n$
(c) attains maximum at exactly two values of $n$
(d) attains maximum for infinitely many values of $n$.

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(a) does not have a maximum
(b) attains maximum at exactly one value of $n$
(c) attains maximum at exactly two values of $n$
(d) attains maximum for infinitely many values of $n$.

For any integer $n \geq 1$, define $a_n = \frac{1000^n}{n!}$. Then the sequence $\{a_n\}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$
(A) does not have a maximum
(B) attains maximum at exactly one value of $n$
(C) attains maximum at exactly two values of $n$
(D) attains maximum for infinitely many values of $n$

The limit $$\lim _ { n \rightarrow \infty } \left( 2 ^ { - 2 ^ { n + 1 } } + 2 ^ { - 2 ^ { n - 1 } } \right) ^ { 2 ^ { - n } }$$ equals
(A) 1.
(B) $\frac { 1 } { \sqrt { 2 } }$.
(C) 0.
(D) $\frac { 1 } { 4 }$.

Let $p , q$ be integers and let $\alpha , \beta$ be the roots of the equation, $x ^ { 2 } - x - 1 = 0$, where $\alpha \neq \beta$. For $n = 0,1,2 , \ldots$, let $a _ { n } = p \alpha ^ { n } + q \beta ^ { n }$.
FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt { 5 } = 0$, then $a = 0 = b$.
$a _ { 12 } =$
[A] $a _ { 11 } - a _ { 10 }$
[B] $a _ { 11 } + a _ { 10 }$
[C] $2 a _ { 11 } + a _ { 10 }$
[D] $a _ { 11 } + 2 a _ { 10 }$

Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - x - 1 = 0$, with $\alpha > \beta$. For all positive integers $n$, define $$\begin{aligned} & a _ { n } = \frac { \alpha ^ { n } - \beta ^ { n } } { \alpha - \beta } , \quad n \geq 1 \\ & b _ { 1 } = 1 \text { and } \quad b _ { n } = a _ { n - 1 } + a _ { n + 1 } , \quad n \geq 2 . \end{aligned}$$ Then which of the following options is/are correct?
(A) $\quad a _ { 1 } + a _ { 2 } + a _ { 3 } + \cdots + a _ { n } = a _ { n + 2 } - 1$ for all $n \geq 1$
(B) $\quad \sum _ { n = 1 } ^ { \infty } \frac { a _ { n } } { 10 ^ { n } } = \frac { 10 } { 89 }$
(C) $b _ { n } = \alpha ^ { n } + \beta ^ { n }$ for all $n \geq 1$
(D) $\quad \sum _ { n = 1 } ^ { \infty } \frac { b _ { n } } { 10 ^ { n } } = \frac { 8 } { 89 }$

Let $\alpha , \beta$ be the distinct roots of the equation $x ^ { 2 } - \left( t ^ { 2 } - 5 t + 6 \right) x + 1 = 0 , t \in \mathbb { R }$ and $a _ { n } = \alpha ^ { n } + \beta ^ { n }$. Then the minimum value of $\frac { a _ { 2023 } + a _ { 2025 } } { a _ { 2024 } }$ is
(1) $- 1 / 4$
(2) $- 1 / 4$
(3) $- 1 / 2$
(4) $1 / 4$

If $\alpha , \beta$ are the roots of the equation, $\mathrm { x } ^ { 2 } - \mathrm { x } - 1 = 0$ and $\mathrm { S } _ { \mathrm { n } } = 2023 \alpha ^ { \mathrm { n } } + 2024 \beta ^ { \mathrm { n } }$, then
(1) $2 \quad \mathrm {~S} _ { 12 } = \mathrm { S } _ { 11 } + \mathrm { S } _ { 10 }$
(2) $\mathrm { S } _ { 12 } = \mathrm { S } _ { 11 } + \mathrm { S } _ { 10 }$
(3) $2 \mathrm {~S} _ { 11 } = \mathrm { S } _ { 12 } + \mathrm { S } _ { 10 }$
(4) $\mathrm { S } _ { 11 } = \mathrm { S } _ { 10 } + \mathrm { S } _ { 12 }$

Let $\alpha , \beta ; \alpha > \beta$, be the roots of the equation $x ^ { 2 } - \sqrt { 2 } x - \sqrt { 3 } = 0$. Let $\mathrm { P } _ { n } = \alpha ^ { n } - \beta ^ { n } , n \in \mathrm {~N}$. Then $( 11 \sqrt { 3 } - 10 \sqrt { 2 } ) \mathrm { P } _ { 10 } + ( 11 \sqrt { 2 } + 10 ) \mathrm { P } _ { 11 } - 11 \mathrm { P } _ { 12 }$ is equal to
(1) $10 \sqrt { 3 } \mathrm { P } _ { 9 }$
(2) $11 \sqrt { 3 } P _ { 9 }$
(3) $10 \sqrt { 2 } \mathrm { P } _ { 9 }$
(4) $11 \sqrt { 2 } \mathrm { P } _ { 9 }$

Q61. Let $\alpha , \beta$ be the distinct roots of the equation $x ^ { 2 } - \left( t ^ { 2 } - 5 t + 6 \right) x + 1 = 0 , t \in \mathbb { R }$ and $a _ { n } = \alpha ^ { n } + \beta ^ { n }$. Then the minimum value of $\frac { a _ { 2023 } + a _ { 2025 } } { a _ { 2024 } }$ is
(1) $- 1 / 4$
(2) $- 1 / 4$
(3) $- 1 / 2$
(4) $1 / 4$

In the following, all terms of the sequences under consideration are real numbers.
(1) For a geometric sequence $\left\{ s _ { n } \right\}$ with first term 1 and common ratio 2,
$$s _ { 1 } s _ { 2 } s _ { 3 } = \square , \quad s _ { 1 } + s _ { 2 } + s _ { 3 } = \square$$
(2) Let $\left\{ s _ { n } \right\}$ be a geometric sequence with first term $x$ and common ratio $r$. Let $a, b$ be real numbers (with $a \neq 0$), and suppose the first three terms of $\left\{ s _ { n } \right\}$ satisfy
$$\begin{aligned} & s _ { 1 } s _ { 2 } s _ { 3 } = a ^ { 3 } \\ & s _ { 1 } + s _ { 2 } + s _ { 3 } = b \end{aligned}$$
Then
$$x r = \square$$
Furthermore, using (2) and (3), we find the relation satisfied by $r, a, b$:
$$\text { エ } r ^ { 2 } + ( \text { オ } - \text { カ } ) r + \text { 倍 } = 0$$
Since there exists a real number $r$ satisfying (4),
$$\text { ク } a ^ { 2 } + \text { ケ } a b - b ^ { 2 } \leqq 0$$
Conversely, when $a, b$ satisfy (5), we can find the values of $r, x$ using (3) and (4).
(3) When $a = 64 , b = 336$, consider the geometric sequence $\left\{ s _ { n } \right\}$ satisfying conditions (1) and (2) in (2) with common ratio greater than 1. Using (3) and (4), we find the common ratio $r$ and first term $x$ of $\left\{ s _ { n } \right\}$: $r = \square , x =$ サシ.
Using $\left\{ s _ { n } \right\}$, define the sequence $\left\{ t _ { n } \right\}$ by
$$t _ { n } = s _ { n } \log _ { \square } s _ { n } \quad ( n = 1,2,3 , \cdots )$$
Then the general term of $\left\{ t _ { n } \right\}$ is $t _ { n } = ( n +$ ス $) \cdot$ コ $^ { n + }$ セ. The sum $U _ { n }$ of the first $n$ terms of $\left\{ t _ { n } \right\}$ is found by computing $U _ { n } - \square U _ { n }$:

Given a real number sequence $\left\langle a _ { n } \right\rangle$ satisfying $a _ { 1 } = 1 , a _ { n + 1 } = \frac { 2 n + 1 } { 2 n - 1 } a _ { n } , n$ is a positive integer. Select the correct options.
(1) $a _ { 2 } = 3$
(2) $a _ { 4 } = 9$
(3) $\left\langle a _ { n } \right\rangle$ is a geometric sequence
(4) $\sum _ { n = 1 } ^ { 20 } a _ { n } = 400$
(5) $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n } = 2$