All Tangled Up

This problem follows on from Twisting and Turning and More Twisting and Turning

In More Twisting and Turning, you were invited to find a strategy to reduce any fraction $\frac ab$ to zero using only the operations Twist ($x \mapsto x+1$) and Turn ($x \mapsto -\frac1x$). In this problem, we will call these operations $T$ for twist, and $R$ for rotate (since 'turn' rather unhelpfully begins with a T too!).

To end up at $\frac{4}{5}$, you can carry out the following sequence of operations: $T,T,T,T,T,R,T$ which could be written more concisely as $T^5RT = \frac45$.

Can you find a sequence of operations that leads to $\frac{9}{10}$?
What about $\frac{23}{24}$?

Can you find a sequence of operations that gets from $0$ to the fraction $\frac{n}{n+1}$?

• $T^2RT$
• $T^2RT^2RT$
• $T^2RT^2RT^2RT$

Can you find other patterns that lead to interesting fractions?

Can you prove that it is possible to start at zero and reach any fraction using only the operations $T$ and $R$?