This exercise is aimed at showing that zero-padding in the time domain interpolates the frequency domain. Since this is homework, I'll give you the beginning of the solution.

Just like you did, start with:

\begin{align}
Y[k] &= \sum_{k=0}^{N-1}x[n]e^{-j2\pi k\frac{n}{2N}}\\
&= \sum_{k=0}^{N-1}x[n]e^{j2\pi k\frac{n}{2N}}e^{-j2\pi k\frac{n}{N}}\\
&= \mathcal{F}\left\{x[n]e^{j2\pi k\frac{n}{2N}}\right\}
\end{align}
where $\mathcal{F}$ denotes the DFT operator.

Next, we know that multiplication in the time domain is convolution in the frequency domain, so:
$$Y[k]  = X[k] * \mathcal{F}\left\{e^{j\pi k\frac{n}{N}}\right\}$$

The next step is to compute $\mathcal{F}\left\{e^{j\pi k\frac{n}{N}}\right\}$  
Can you take it from there?