Similarity theorem in Fourier analysis
If you have a $T$-periodic function $g(t)$ with Fourier coefficients $c_n$

$$g(t)=\sum_{n=-\infty}^{\infty}c_ne^{in\omega_0t},\quad\omega_0=\frac{2\pi}{T}$$

and you create a new function $f(t)$ with Fourier coefficients $d_n$ defined by

$$d_n=c_{n/2},\quad n\textrm{ even}\\
d_n=0,\quad n\textrm{ odd}$$

i.e. the spectrum is expanded by a factor two by inserting zeros between the original Fourier coefficients. Then $f(t)$ is given by

$$f(t)=\sum_{n=-\infty}^{\infty}d_ne^{in\omega_0t}=
\sum_{n\textrm{ even}}c_{n/2}e^{in\omega_0t}=
\sum_{n=-\infty}^{\infty}c_ne^{i2n\omega_0t}=g(2t)$$

So you get a compressed time domain signal with period $T/2$, as directly shown in the first equation of the problem formulation in your question.