Exercise:
$Proof$.
Base case: When $n=1$, we have $4^n=4^1=4$, and if any corner is removed, we can cover the three other subtriangles by one trapezoidal tile.
Induction step: Suppose the statement holds for $n$, and that an equilateral triangle is cut into $4^{n+1}$ congruent triangles, and one corner is removed. Now $4^{n+1}=4\cdot 4^n$, so we can think about the triangle as $4$ subtriangles, each cut into $4^n$ smaller triangles. Assume without loss of generality that the left corner was removed, like so
Now place a trapezoidal on the right side, so that it covers a corner of each of the three subtriangles, i.e,
Then by inductive hypothesis, we can cover each subtriangle with the given trapezoidal tiles, which then covers the entire triangle. $\square$
Is this proof valid? Im concerned about the pictures that I used since they pretty much assume $n=2$, but it clearly works for other sizes aswell, the black triangle would just be smaller, and you can similarly place the trapezoidal on the right side so that it covers a corner of the other subtriangles.