I need to show the following statement:
Show that for every Frenet curve $c:I\to\mathbb{R}^n$, the curvatures $\kappa_1(t),\ldots,\kappa_{n-1}(t)$ satisfy the following equality:
$$\prod_{i=1}^{n-1}(\kappa_i(t))^{n-i}=\frac{\det(\dot{c}(t),\ldots,c^{(n)}(t))}{\|\dot{c}(t)\|^{n(n+1)/2}}.$$
My Idea: Show it by induction over $n$.
My solution: Let $n=2$. Then:
$$\prod_{i=1}^{2-1}\kappa_i(t)^{2-i}= \kappa_1(t) = \frac{\det(\dot{c}(t),\ddot{c}(t))}{\|\dot{c}(t)\|^3}. \checkmark $$
Now, $n-1\mapsto n$:
\begin{align*} \prod_{i=1}^n (\kappa_i(t))^{n+1-i} &= \prod_{i=1}^n (\kappa_i(t))^{n-i}\cdot\kappa_i(t) \\ &= \kappa_n(t)\cdot\prod_{i=1}^{n-1} \kappa_i(t) \prod_{i=1}^{n-1}\kappa_i(t)^{n-i} \\ &=\prod_{i=1}^n \kappa_i(t)\cdot\frac{\det(\dot{c}(t),\ldots,c^{(n)}(t))}{\|\dot{c}(t)\|^{n(n+1)/2}} \end{align*}
What is $\prod_{i=1}^n \kappa_i(t)$?
Hint: You need to write down the Frenet equations (which actually define the $\kappa_i$) for a curve parametrized by arclength, and then use the product rule to compute the derivatives of $c$. Then use the chain rule to adjust for non-unit-speed curves.