1) Let $\displaystyle f_{k}$ denote thekth Fibonacci number. Find the simple continued fraction, terminating with the partial quotient of 1, of $\displaystyle f_{k+1}/f_{k}$, wherekis a positive integer.

2) Show that if $\displaystyle a_{0} > 0$, then

$\displaystyle p_{k}/p_{k-1} = [a_{k}; a_{k-1},.....,a_{1}, a_{0}]$

and

$\displaystyle q_{k}/q_{k-1} = [a_{k}; a_{k-1},.....,a_{2}, a_{1}]$,

where $\displaystyle C_{k-1} = p_{k-1}/q_{k-1}$ and $\displaystyle C_{k} = p_{k}/q_{k}, k \geq 1$, are successive convergents of the continued fraction $\displaystyle [a_{0};a_{1},...,a_{n}]$

(Given a hint for this one: Use the relation $\displaystyle p_{k} = a_{k}p_{k-1}+p_{k-2}$ to show that $\displaystyle p_{k}/p_{k-1} = a_{k} + 1/(P_{k-1}/p_{k-2}$).)