Use Fourier transform to find Fourier series coeficcients I understand that the Fourier Transform can be seen as a generalisation of the Fourier Series, where the period $T_0 \to \infty$ . Now I have encountered this strange question (in an engineering course on signal analysis):

Given a periodic function $x(t)$, find the Fourier Series coefficients $X_n$ by using the Fourier Transform.

What does this mean? How can it be done? As I see it, FS and FT are similar concepts, but they are not the same operation.
For reference, $x(t) = rect(\frac{t-0.25}{0.25}) * \Delta _1 (t)$ but I am seeking an answer in terms of any periodic function $x(t)$ .

You question is about the relation of the two fourier transforms, on $\mathbb R$ and on $\mathbb T = \mathbb R / 2\pi\mathbb Z$. The first fourier transform can be given for $f\in L^1(\mathbb R)$ as
$$\mathcal F f(\xi) = \frac1{\sqrt{2\pi}} \int_{\mathbb R} f(x) e^{-i\xi x}dx$$
While the other maps a $2\pi$-periodic function $g:\mathbb T \to \mathbb R$ by
$$\mathcal F g(k) = \frac1{2\pi} \int_{\mathbb T} g(x) e^{-ikx} dx$$
Now the relation asked for is the following identity. Let $g$ be a $2\pi$-periodic function and $f\in\mathcal S(\mathbb R)$ a schwartz-function satisfying
$$g(x) = \sum_{k\in\mathbb Z} f(x+2\pi k)$$
Then
$$\mathcal F g(k) = \mathcal F f(k) \qquad \forall k\in\mathbb Z$$
Where the first FT is on $\mathbb T$ and the second is on $\mathbb R$.
However, the terms fourier transform and fourier series can be used synonymously if the domain of the fourier transform is clear ($\mathbb T$). Note that other periods than $2\pi$ can be deduced from this and similar equalities hold.
