<s>[INST] Solving for the minimal number of moves required in the Tower of Hanoi According to ProofWiki,

Thus we arrive at our recurrence rule: $T_{n} = 2T_{n - 1} + 1$.

The solution when solving this recurrence relation in that page is a proof by induction. Can someone verify if my attempt is correct?

The given equations are $T_{n} = 2T_{n - 1} + 1$ $[1]$, $T_{0} = 0$, and $T_{1} = 1$. By algebraic manipulations,
\begin{align*}T_{n} &= 2T_{n - 1} + 1 \\ T_{n} + 1&= 2T_{n - 1} + 2 \\ T_{n} + 1&= 2(T_{n - 1} + 1) \end{align*}
Let $u_{n} = T_{n} + 1$ $[2]$. Then, the last equation simplifies to $u_{n} = 2u_{n - 1}$. Then,
\begin{align*}u_{n} &= 2u_{n - 1} \\ \frac{u_{n}}{u_{n - 1}} &= 2\end{align*}
Notice that this recurrence relation forms a geometric progression with common ratio of $2$. Multiplying the equation to itself but with shifted indices from $n$ to $1$, we get $n - 1 + 1 = 1$ terms.
\begin{align*}\left(\frac{u_{n}}{u_{n - 1}}\right)\left(\frac{u_{n - 1}}{n - 2}\right)\left(\frac{u_{n - 2}}{u_{n - 3}}\right)\cdots\left(\frac{u_{2}}{u_{1}}\right)\left(\frac{u_{1}}{u_{0}}\right)&=2^{n} \\ \frac{u_{n}}{u_{0}} &= 2^{n} \\ u_{n} &= 2^{n}u_{0}\qquad[3]\end{align*}
Solving for $u_{0}$ when $T_{0} = 0$ from $[2]$,
\begin{align*}u_{n} &= T_{n} + 1 \\ u_{0} &= T_{0} + 1 \\ u_{0} &= 0 + 1 \\ u_{0} &= 1\end{align*}
By substitution in $[3]$,
\begin{align*}u_{n} &= 2^{n}(1) \\ u_{n} &= 2^{n} \\\\ T_{n} + 1 &= 2^{n} \\ T_{n} &= 2^{n} - 1 \qquad\qquad \square\end{align*}

Edit (clarification): Can someone verify my attempt? If it's correct, can it be simplified further?
 [/INST] To remove this question from the Unanswered tab, I will post an CW answer.

As Brian M. Scott commented,

Some of the computational steps. I’d be perfectly happy, for instance, if you jumped straight from $u_n=2u_n−1$ and $u_0=1$ to $u_n=2^n$, simply noting that this is clearly a geometric progression with ratio 2 and 0-th term 1. Thus, $$T_n=u_n−1=2^n−1,$$
and you’re done. If this material is still very new to you, there is nothing wrong with filling in more details in order to feel more comfortable, but most of the ones that you included really can be omitted in most contexts.

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