Q1 We are to find the general term $a _ { n }$ of the sequence $\left\{ a _ { n } \right\}$ which is determined by the recurrence formula
$$a _ { 1 } = 18 , \quad a _ { n + 1 } - 12 a _ { n } + 3 ^ { n + 2 } = 0 \quad ( n = 1,2,3 , \cdots ) .$$
When we define a sequence $\left\{ b _ { n } \right\}$ by
$$b _ { n } = \frac { a _ { n } } { \mathbf{A}^n } \quad ( n = 1,2,3 , \cdots ) ,$$
$\left\{ b _ { n } \right\}$ satisfies
$$b _ { 1 } = \mathbf { B } , \quad b _ { n + 1 } - \mathbf { C } \; b _ { n } + \mathbf { D } = 0 \quad ( n = 1,2,3 , \cdots ) .$$
This recurrence formula can be transformed into
$$b _ { n + 1 } - \mathbf { E } = \mathbf{F} ( b_n - \mathbf{E} )$$
Next, when we define a sequence $\left\{ c _ { n } \right\}$ by
$$c _ { n } = b _ { n } - \mathbf { E } \quad ( n = 1,2,3 , \cdots ) ,$$
$\left\{ c _ { n } \right\}$ is a geometric progression such that the first term is $\mathbf{G}$ and the common ratio is $\mathbf{H}$.
Hence we have
$$a _ { n } = \mathbf { I } ^ { n } \left( \mathbf { J } \cdot \mathbf { K } ^ { n - 1 } + \mathbf { L } \right) \quad ( n = 1,2,3 , \cdots ) .$$

The sequence $\left\{ a _ { n } \right\}$ is defined by
$$a _ { 1 } = \frac { 2 } { 9 } , \quad a _ { n } = \frac { ( n + 1 ) ( 2 n - 3 ) } { 3 n ( 2 n + 1 ) } a _ { n - 1 } \quad ( n = 2,3,4 , \cdots ) .$$
We are to find the general term $a _ { n }$ and the infinite sum $\sum _ { n = 1 } ^ { \infty } a _ { n }$.
(1) For A $\sim$ E in the following sentences, choose the correct answers from among (0) $\sim$ (9) below.
First, when we set $b _ { n } = \frac { n + 1 } { 3 ^ { n } a _ { n } }$ and express $\frac { b _ { n } } { b _ { n - 1 } }$ in terms of $n$, we have
$$\frac { b _ { n } } { b _ { n - 1 } } = \frac { \mathbf { A } } { \mathbf { B } } \cdot \frac { a _ { n - 1 } } { a _ { n } } = \frac { \mathbf { C } } { \mathbf { D } }$$
From this equation, we have
$$a _ { n } = \frac { n + 1 } { 3 ^ { n } ( \mathbf { E } ) ( 2 n + 1 ) } .$$
(0) $n - 1$
(1) $n$
(2) $n + 1$
(3) $2 n - 1$
(4) $2 n + 1$
(5) $2 n - 3$ (6) $2 n + 3$ (7) $3 n - 1$ (8) $3 n$ (9) $3 n + 1$
(2) Next, let $c _ { n } = \frac { 1 } { 3 ^ { n } ( 2 n + 1 ) } ( n = 0,1,2 , \cdots )$. When we set $a _ { n } = A c _ { n - 1 } + B c _ { n }$, we see that $A = \frac { \mathbf { F } } { \mathbf { G } }$ and $B = \frac { \mathbf { H I } } { \mathbf { G } }$. Using this result to find $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, we have
$$S _ { n } = \frac { \mathbf { K } } { \mathbf { L } } \left( \mathbf { M } \right)$$
Hence we obtain
$$\sum _ { n = 1 } ^ { \infty } a _ { n } = \lim _ { n \rightarrow \infty } S _ { n } = \frac { \mathbf { N } } { \mathbf { O } }$$

Let $\{a_n\}$ be a sequence such that the sum $S_n$ of the terms from the first term to the $n$-th term is $$S_n = \frac{n^2 - 17n}{4},$$ and let $\{b_n\}$ be the sequence defined by $$b_n = a_n \cdot a_{n+5} \quad (n = 1, 2, 3, \cdots)$$
(1) For $\mathbf{A}$ $\sim$ $\mathbf{C}$ in the following sentences, choose the correct answer from among choices (0) $\sim$ (9) below.
Let us find the sum $T_n$ of the terms of sequence $\{b_n\}$ from the first term to the $n$-th term.
Since $a_n = \mathbf{A}$, we have $b_n = \mathbf{B}$. Hence we obtain $$T_n = \mathbf{C}.$$
(0) $\frac{n-7}{2}$
(1) $\frac{n-9}{2}$
(2) $\frac{n-11}{2}$
(3) $\frac{n^2 - 12n + 27}{4}$
(4) $\frac{n^2 - 13n + 36}{4}$
(5) $\frac{n^2 - 14n + 45}{4}$ (6) $\frac{n(n^2 - 17n + 83)}{12}$ (7) $\frac{n(n^2 - 17n + 89)}{12}$ (8) $\frac{n(n^2 - 18n + 83)}{12}$ (9) $\frac{n(n^2 - 18n + 89)}{12}$
(2) Next, let us find the minimum value of $T_n$.
When $n \leqq \mathbf{D}$ or $\mathbf{EF} \leqq n$, we see that $b_n > 0$. On the other hand, when $\mathbf{G} \leqq n \leqq \mathbf{H}$, we see that $b_n < 0$.
Hence $T_n$ is minimized at $n = \mathbf{I}$, $n = \mathbf{J}$ and $n = \mathbf{K}$, and its minimum value is $\mathbf{L}$. (Answer in the order such that $\mathbf{I} < \mathbf{J} < \mathbf{K}$.)

2. For ALL APPLICANTS.

(i) Find a pair of positive integers, $x _ { 1 }$ and $y _ { 1 }$, that solve the equation
$$\left( x _ { 1 } \right) ^ { 2 } - 2 \left( y _ { 1 } \right) ^ { 2 } = 1$$
(ii) Given integers $a , b$, we define two sequences $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ and $y _ { 1 } , y _ { 2 } , y _ { 3 } , \ldots$ by setting
$$x _ { n + 1 } = 3 x _ { n } + 4 y _ { n } , \quad y _ { n + 1 } = a x _ { n } + b y _ { n } , \quad \text { for } n \geqslant 1$$
Find positive values for $a , b$ such that
$$\left( x _ { n + 1 } \right) ^ { 2 } - 2 \left( y _ { n + 1 } \right) ^ { 2 } = \left( x _ { n } \right) ^ { 2 } - 2 \left( y _ { n } \right) ^ { 2 } .$$
(iii) Find a pair of integers $X , Y$ which satisfy $X ^ { 2 } - 2 Y ^ { 2 } = 1$ such that $X > Y > 50$.
(iv) (Using the values of $a$ and $b$ found in part (ii)) what is the approximate value of $x _ { n } / y _ { n }$ as $n$ increases?

2. For ALL APPLICANTS.

A list of real numbers $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is defined by $x _ { 1 } = 1 , x _ { 2 } = 3$ and then for $n \geqslant 3$ by
$$x _ { n } = 2 x _ { n - 1 } - x _ { n - 2 } + 1$$
So, for example,
$$x _ { 3 } = 2 x _ { 2 } - x _ { 1 } + 1 = 2 \times 3 - 1 + 1 = 6$$
(i) Find the values of $x _ { 4 }$ and $x _ { 5 }$.
(ii) Find values of real constants $A , B , C$ such that for $n = 1,2,3$,
$$x _ { n } = A + B n + C n ^ { 2 }$$
(iii) Assuming that equation ( $*$ ) holds true for all $n \geqslant 1$, find the smallest $n$ such that $x _ { n } \geqslant 800$.
(iv) A second list of real numbers $y _ { 1 } , y _ { 2 } , y _ { 3 } , \ldots$ is defined by $y _ { 1 } = 1$ and
$$y _ { n } = y _ { n - 1 } + 2 n$$
Find, explaining your reasoning, a formula for $y _ { n }$ which holds for $n \geqslant 2$. What is the approximate value of $x _ { n } / y _ { n }$ for large values of $n$ ?

2. For ALL APPLICANTS.

Suppose that $x$ satisfies the equation
$$x ^ { 3 } = 2 x + 1$$
(i) Show that
$$x ^ { 4 } = x + 2 x ^ { 2 } \quad \text { and } \quad x ^ { 5 } = 2 + 4 x + x ^ { 2 } .$$
(ii) For every integer $k \geqslant 0$, we can uniquely write
$$x ^ { k } = A _ { k } + B _ { k } x + C _ { k } x ^ { 2 }$$
where $A _ { k } , B _ { k } , C _ { k }$ are integers. So, in part (i), it was shown that
$$A _ { 4 } = 0 , B _ { 4 } = 1 , C _ { 4 } = 2 \quad \text { and } \quad A _ { 5 } = 2 , B _ { 5 } = 4 , C _ { 5 } = 1 .$$
Show that
$$A _ { k + 1 } = C _ { k } , \quad B _ { k + 1 } = A _ { k } + 2 C _ { k } , \quad C _ { k + 1 } = B _ { k }$$
(iii) Let
$$D _ { k } = A _ { k } + C _ { k } - B _ { k }$$
Show that $D _ { k + 1 } = - D _ { k }$ and hence that
$$A _ { k } + C _ { k } = B _ { k } + ( - 1 ) ^ { k }$$
(iv) Let $F _ { k } = A _ { k + 1 } + C _ { k + 1 }$. Show that
$$F _ { k } + F _ { k + 1 } = F _ { k + 2 }$$

5. For ALL APPLICANTS.

This question concerns the sum $s _ { n }$ defined by
$$s _ { n } = 2 + 8 + 24 + \cdots + n 2 ^ { n }$$
(i) Let $f ( n ) = ( A n + B ) 2 ^ { n } + C$ for constants $A , B$ and $C$ yet to be determined, and suppose $s _ { n } = f ( n )$ for all $n \geqslant 1$. By setting $n = 1,2,3$, find three equations that must be satisfied by $A , B$ and $C$.
(ii) Solve the equations from part (i) to obtain values for $A , B$ and $C$.
(iii) Using these values, show that if $s _ { k } = f ( k )$ for some $k \geqslant 1$ then $s _ { k + 1 } = f ( k + 1 )$.
You may now assume that $f ( n ) = s _ { n }$ for all $n \geqslant 1$.
(iv) Find simplified expressions for the following sums:
$$\begin{aligned} & t _ { n } = n + 2 ( n - 1 ) + 4 ( n - 2 ) + 8 ( n - 3 ) + \cdots + 2 ^ { n - 1 } 1 , \\ & u _ { n } = \frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } . \end{aligned}$$
(v) Find the sum
$$\sum _ { k = 1 } ^ { n } s _ { k }$$
If you require additional space please use the pages at the end of the booklet

2. For ALL APPLICANTS.

There is a unique real number $\alpha$ that satisfies the equation
$$\alpha ^ { 3 } + \alpha ^ { 2 } = 1$$
[You are not asked to prove this.]
(i) Show that $0 < \alpha < 1$.
(ii) Show that
$$\alpha ^ { 4 } = - 1 + \alpha + \alpha ^ { 2 }$$
(iii) Four functions of $\alpha$ are given in (a) to (d) below. In a similar manner to part (ii), each is equal to a quadratic expression
$$A + B \alpha + C \alpha ^ { 2 }$$
in $\alpha$, where $A , B , C$ are integers. (So in (ii) we found $A = - 1 , B = 1 , C = 1$.) You may assume in each case that the quadratic expression is unique.
In each case below find the quadratic expression in $\alpha$.
(a) $\alpha ^ { - 1 }$.
(b) The infinite sum
$$1 - \alpha + \alpha ^ { 2 } - \alpha ^ { 3 } + \alpha ^ { 4 } - \alpha ^ { 5 } + \cdots$$
(c) $( 1 - \alpha ) ^ { - 1 }$.
(d) The infinite product
$$( 1 + \alpha ) \left( 1 + \alpha ^ { 2 } \right) \left( 1 + \alpha ^ { 4 } \right) \left( 1 + \alpha ^ { 8 } \right) \left( 1 + \alpha ^ { 16 } \right) \cdots$$

2. For ALL APPLICANTS.

The functions $f ( n )$ and $g ( n )$ are defined for positive integers $n$ as follows:
$$f ( n ) = 2 n + 1 , \quad g ( n ) = 4 n$$
This question is about the set $S$ of positive integers that can be achieved by applying, in some order, a combination of $f _ { \mathrm { s } }$ and $g \mathrm {~s}$ to the number 1 . For example as
$$g f g ( 1 ) = g f ( 4 ) = g ( 9 ) = 36$$
and
$$f f g g ( 1 ) = f f g ( 4 ) = f f ( 16 ) = f ( 33 ) = 67$$
then both 36 and 67 are in $S$.
(i) Write out the binary expansion of 100 (one hundred). [0pt] [Recall that binary is base 2. Every positive integer $n$ can be uniquely written as a sum of powers of 2, where a given power of 2 can appear no more than once. So, for example, $13 = 2 ^ { 3 } + 2 ^ { 2 } + 2 ^ { 0 }$ and the binary expansion of 13 is 1101 .]
(ii) Show that 100 is in $S$ by describing explicitly a combination of $f _ { \mathrm { S } }$ and $g$ s that achieves 100 .
(iii) Show that 200 is not in $S$.
(iv) Show that, if $n$ is in $S$, then there is only one combination of applying $f _ { \mathrm { s } }$ and $g$ s in order to achieve $n$. (So, for example, 67 can only be achieved by applying $g$ then $g$ then $f$ then $f$ in that order.)
(v) Let $u _ { k }$ be the number of elements $n$ of $S$ that lie in the range $2 ^ { k } \leqslant n < 2 ^ { k + 1 }$. Show that
$$u _ { k + 2 } = u _ { k + 1 } + u _ { k }$$
for $k \geqslant 0$.
(vi) Let $s _ { k }$ be the number of elements $n$ of $S$ that lie in the range $1 \leqslant n < 2 ^ { k + 1 }$. Show that
$$s _ { k + 2 } = s _ { k + 1 } + s _ { k } + 1$$
for $k \geqslant 0$.
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2. For ALL APPLICANTS.

For $n$ a positive whole number, and for $x \neq 0$, let $p _ { n } ( x ) = x ^ { n } + x ^ { - n }$. [0pt] (i) [3 marks] Sketch the graph of $y = p _ { 1 } ( x )$. Label any turning points on your sketch.
(ii) $[ 1$ mark $]$ Show that $p _ { 2 } ( x ) = p _ { 1 } ( x ) ^ { 2 } - 2$.
(iii) $[ 1$ mark $]$ Find an expression for $p _ { 3 } ( x )$ in terms of $p _ { 1 } ( x )$. [0pt] (iv) [5 marks] Find all real solutions $x$ to the equation
$$x ^ { 4 } + x ^ { 3 } - 10 x ^ { 2 } + x + 1 = 0$$
(v) [5 marks] Find all real solutions $x$ to the equation
$$x ^ { 7 } + 2 x ^ { 6 } - 5 x ^ { 5 } - 7 x ^ { 4 } + 7 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1 = 0 .$$

5. For ALL APPLICANTS.

Define the sequence, $F _ { n }$, as follows: $F _ { 1 } = 1 , F _ { 2 } = 1$, and for $n \geqslant 3$,
$$F _ { n } = F _ { n - 1 } + F _ { n - 2 } .$$
(i) [3 marks] What are the values $F _ { 3 } , F _ { 4 } , F _ { 5 }$ ? [0pt] (ii) [1 mark] Using the equation (*) repeatedly, in terms of $n$, how many additions do you need to calculate $F _ { n }$ ?
We now consider sequences of 0 's and 1 's of length $n$, that do not have two consecutive 1 's. So, for $n = 5$, for example, ( $0,1,0,0,1$ ) and ( $1,0,1,0,1$ ) would be valid sequences, but ( $0,1,1,0,0$ ) would not. Let $S _ { n }$ denote the number of valid sequences of length $n$. [0pt] (iii) [1 mark] What are $S _ { 1 }$ and $S _ { 2 }$ ? [0pt] (iv) [3 marks] For $n \geqslant 3$, by considering the first element of the sequence of 0 's and 1's, show that $S _ { n }$ satisfies the same equation (*). Hence conclude that $S _ { n } = F _ { n + 2 }$ for all $n$. [0pt] (v) [2 marks] For $n \geqslant 2$, by considering valid sequences of length $2 n - 3$ and focusing on the element in the $( n - 1 ) ^ { \text {th } }$ position, show that,
$$F _ { 2 n - 1 } = F _ { n } ^ { 2 } + F _ { n - 1 } ^ { 2 }$$
(vi) [3 marks] For $n \geqslant 2$, show that,
$$F _ { 2 n } = F _ { n } ^ { 2 } + 2 F _ { n } F _ { n - 1 }$$
(vii) [2 marks] Let $k \geqslant 3$ be an integer. By using the equations (O) and (E) repeatedly, how many arithmetic operations do you need to calculate $F _ { 2 ^ { k } }$ ? You should only count additions and multiplications needed to calculate values using the equations (O) and (E) .

10. Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a sequence of real numbers satisfying $a_{n+1} = \frac{n(n+1)}{2} - a_{n}$ for all positive integers $n$. Which of the following options are correct?
(1) If $a_{1} = 1$, then $a_{2} = 1$
(2) If $a_{1}$ is an integer, then every term of the sequence is an integer
(3) If $a_{1}$ is irrational, then every term of the sequence is irrational
(4) $a_{2} \leq a_{4} \leq \cdots \leq a_{2n} \leq \cdots$ (where $n$ is a positive integer)
(5) If $a_{k}$ is odd, then $a_{k+2}, a_{k+4}, \ldots, a_{k+2n}, \ldots$ are all odd (where $n$ is a positive integer)

For a real number $a$, let $[a]$ denote the greatest integer not exceeding $a$. For example: $[1.2] = [\sqrt{2}] = 1$, $[-1.2] = -2$. Consider the irrational number $\theta = \sqrt{10001}$. Select the correct options.
(1) $a - 1 < [a] \leq a$ holds for all real numbers $a$
(2) The sequence $b_{n} = \frac{[n\theta]}{n}$ diverges, where $n$ is a positive integer
(3) The sequence $c_{n} = \frac{[-n\theta]}{n}$ diverges, where $n$ is a positive integer
(4) The sequence $d_{n} = n\left[\frac{\theta}{n}\right]$ diverges, where $n$ is a positive integer
(5) The sequence $e_{n} = n\left[\frac{-\theta}{n}\right]$ diverges, where $n$ is a positive integer

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) $\sum _ { n = 1 } ^ { 100 } a _ { n } > 9$

Find the value of the limit $\lim _ { n \rightarrow \infty } \frac { 10 ^ { 10 } } { n ^ { 10 } } \left[ 1 ^ { 9 } + 2 ^ { 9 } + 3 ^ { 9 } + \cdots + ( 2 n ) ^ { 9 } \right]$ .
(1) $10 ^ { 9 }$
(2) $10 ^ { 9 } \times \left( 2 ^ { 10 } - 1 \right)$
(3) $2 ^ { 9 } \times \left( 10 ^ { 10 } - 1 \right)$
(4) $10 ^ { 9 } \times 2 ^ { 10 }$
(5) $2 ^ { 9 } \times 10 ^ { 10 }$

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) $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n } = 2$

Let $a_1, a_2, a_3, a_4$ be a geometric sequence with first term 10 and common ratio 10. Let $b = \sum_{n=1}^{3} \log_{a_n} a_{n+1}$. Select the correct option.
(1) $2 < b \leq 3$
(2) $3 < b \leq 4$
(3) $4 < b \leq 5$
(4) $5 < b \leq 6$
(5) $6 < b \leq 7$

Suppose two sequences $\langle a_n \rangle$ and $\langle b_n \rangle$ satisfy $b_n + \frac{4n-1}{n} < a_n < 3b_n$ for all positive integers $n$. Given that $\lim_{n \to \infty} a_n = 6$, select the correct options.
(1) $b_n < 6 - \frac{4n-1}{n}$
(2) $b_n > \frac{4n-1}{2n}$
(3) The sequence $\langle b_n \rangle$ may diverge
(4) $a_{10000} < 6.1$
(5) $a_{10000} > 5.9$

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be a geometric sequence with first term 3 and common ratio $3\sqrt{3}$. Select the number of terms $n$ that satisfy the inequality
$$\log_{3} a_{1} - \log_{3} a_{2} + \log_{3} a_{3} - \log_{3} a_{4} + \ldots + (-1)^{n+1} \log_{3} a_{n} > 18$$
among the possible options.
(1) 23
(2) 24
(3) 25
(4) 26
(5) 27

There is a real number sequence $\left\langle a_{n} \right\rangle$, where $a_{n} = \cos\left(n\pi - \frac{\pi}{6}\right)$, and $n$ is a positive integer. Select the correct options.
(1) $a_{1} = -\frac{1}{2}$
(2) $a_{2} = a_{3}$
(3) $a_{4} = a_{24}$
(4) $\left\langle a_{n} \right\rangle$ is a convergent sequence, and $\lim_{n \rightarrow \infty} a_{n} < 1$
(5) $\sum_{n=1}^{\infty} \left(a_{n}\right)^{n} = 3 - 2\sqrt{3}$

A sequence $< a _ { n } >$ satisfies $3 a _ { n + 1 } = a _ { n } + n$ (for all positive integers $n$) and $a _ { 1 } = 2$. Let the sequence $< b _ { n } >$ satisfy $b _ { n } = a _ { n } - \frac { n } { 2 } + \frac { 3 } { 4 }$. Select the correct options.
(1) $a _ { 2 } = 2$
(2) $b _ { 2 } = \frac { 3 } { 4 }$
(3) The sequence $< b _ { n } >$ is a geometric sequence with common ratio $\frac { 2 } { 3 }$
(4) For any positive integer $n$, $3 ^ { n } a _ { n }$ is always a positive integer
(5) $b _ { 10 } < 10 ^ { - 4 }$

16. The sequence $a _ { n }$ is given by the rule:
$$\begin{aligned} a _ { 1 } & = 2 \\ a _ { n + 1 } & = a _ { n } + ( - 1 ) ^ { n } \text { for } n \geq 1 \end{aligned}$$
What is
$$\sum _ { n = 1 } ^ { 100 } a _ { n }$$
A 150
B 250
C - 4750
D 5150
E $\quad 4 \left( 1 - \left( \frac { 1 } { 2 } \right) ^ { 100 } \right)$ F $\quad 4 \left( \left( \frac { 3 } { 2 } \right) ^ { 100 } - 1 \right)$

The function $f$ is defined on the positive integers as follows:
$$f ( 1 ) = 5 , \text { and for } n \geqslant 1 : \quad \begin{array} { l l } f ( n + 1 ) = 3 f ( n ) + 1 & \text { if } f ( n ) \text { is odd } \\ & f ( n + 1 ) = \frac { 1 } { 2 } f ( n ) \end{array} \text { if } f ( n ) \text { is even }$$
The function $g$ is defined on the positive integers as follows:
$$\begin{array} { l l } g ( 1 ) = 3 , \text { and for } n \geqslant 1 : \quad & g ( n + 1 ) = g ( n ) + 5 \\ & \text { if } g ( n ) \text { is odd } \\ g ( n + 1 ) = \frac { 1 } { 2 } g ( n ) & \text { if } g ( n ) \text { is even } \end{array}$$
What is the value of $f ( 1000 ) - g ( 1000 )$ ?
A - 6
B - 5
C 1
D 2
E 4
F 8

The sequence $x_n$ is given by:
$$\begin{aligned} x_1 &= 10 \\ x_{n+1} &= \sqrt{x_n} \text{ for } n \geq 1 \end{aligned}$$
What is the value of $x_{100}$?
[Note that $a^{b^c}$ means $a^{(b^c)}$]

The first seven terms of a sequence of positive integers are:
$$\begin{aligned} & u _ { 1 } = 15 \\ & u _ { 2 } = 21 \\ & u _ { 3 } = 30 \\ & u _ { 4 } = 37 \\ & u _ { 5 } = 44 \\ & u _ { 6 } = 51 \\ & u _ { 7 } = 59 \end{aligned}$$
Consider the following statement about this sequence: (*) If $n$ is a prime number, then $u _ { n }$ is a multiple of 3 or $u _ { n }$ is a multiple of 5 .
What is the smallest value of $n$ that provides a counterexample to $( * )$ ?
A 1
B 2
C 3
D 4
E 5 F 6 G 7