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 $$F _ { k } + F _ { k + 1 } = F _ { k + 2 }$$
(i) [2 marks] Multiplying $x ^ { 3 } = 2 x + 1$ by $x$ we get the first equation. Multiplying by $x$ again we get
\section*{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
$$F _ { k } + F _ { k + 1 } = F _ { k + 2 }$$