Question 3
For two fractions $a/b$ and $c/d$, define their pirate sum by
$$
\frac{a}{b}\diamond\frac{c}{d}=\frac{a+c}{b+d},
$$
where both input fractions are written in lowest terms (that is, simplified as far as possible), and the result is then also reduced to lowest terms. For example, the pirate sum of $2/7$ and $4/5$ is $1/2$.
Let $n\geq3$ be an integer. Initially, the fractions
$$
\frac11,\quad\frac12,\quad\frac13,\quad\ldots,\quad\frac1n
$$
are written on a blackboard. At each move, choose two fractions on the board, erase them, and replace them with their pirate sum. Continue until only one fraction remains.
Determine, as functions of $n$, the largest and the smallest possible values of that final fraction.
$$
\frac{a}{b}\diamond\frac{c}{d}=\frac{a+c}{b+d},
$$
where both input fractions are written in lowest terms (that is, simplified as far as possible), and the result is then also reduced to lowest terms. For example, the pirate sum of $2/7$ and $4/5$ is $1/2$.
Let $n\geq3$ be an integer. Initially, the fractions
$$
\frac11,\quad\frac12,\quad\frac13,\quad\ldots,\quad\frac1n
$$
are written on a blackboard. At each move, choose two fractions on the board, erase them, and replace them with their pirate sum. Continue until only one fraction remains.
Determine, as functions of $n$, the largest and the smallest possible values of that final fraction.