(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?
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 $$F _ { k } + F _ { k + 1 } = F _ { k + 2 }$$
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) .
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 terms $x _ { n }$ of a sequence follow the rule $$x _ { n + 1 } = \frac { x _ { n } + p } { x _ { n } + q }$$ where $p$ and $q$ are real numbers. Given that $x _ { 1 } = 3 , x _ { 2 } = 5$, and $x _ { 3 } = 7$, find the value of $x _ { 4 }$
The function f satisfies the equation $$f ( n ) = 2 \cdot f ( n - 1 ) + 1$$ for integers $n \geq 1$. Given that $f ( 0 ) = 1$, what is $f ( 2 )$? A) 8 B) 7 C) 6 D) 5 E) 4
The sequence $(a _ { k })$ is defined as $$\begin{aligned}
& a _ { 1 } = 40 \\
& a _ { k + 1 } = a _ { k } - k \quad ( k = 1,2,3 , \ldots )
\end{aligned}$$ Accordingly, what is the term $\mathrm { a } _ { 8 }$? A) 4 B) 7 C) 12 D) 15 E) 19
Let $a _ { 1 } , a _ { 2 }$ be real numbers. The sequence $\left( a _ { n } \right)$ satisfies the relation $$a _ { n + 2 } = a _ { n + 1 } + a _ { n } \quad ( n = 1,2 , \cdots )$$ Given that $a _ { 8 } = 6$, what is the sum $a _ { 6 } + a _ { 9 }$? A) 9 B) 10 C) 12 D) 15 E) 16
The sequence $(a_n)$ of real numbers satisfies for every positive integer $n$ $$a_{n+1} = a_n + \frac{(-1)^n \cdot a_n}{2}$$ the equality. If $a_5 = 18$, what is $a_1$? A) 4 B) 8 C) 16 D) 32 E) 64
The sequence $(a_n)$ of real numbers satisfies for every positive integer $n$ $$a_{n} + (-1)^{n} \cdot a_{n+1} = 2^{n}$$ If $a_{1} = 0$, what is the sum $a_{3} + a_{4} + a_{5} + a_{6}$? A) 6 B) 8 C) 12 D) 16 E) 20