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 $n$ ?
(i) [2 marks] Using the given recurrence relation
\section*{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 $n$ ?