Fourier coefficients of sum of two functions with different fundamental periods?
Since this looks like a homework and solving people's homework is outside the scope of this group, here are some hints

To have a Fourier Series, a function needs to be periodic.
$$ g(t)=g(t+T_1)  \quad  \Rightarrow  \sum_k \left(a_k \sin(2\pi k \frac{t}{T_1}) + b_k \cos (2\pi k \frac{t}{T_1}) \right) $$ 
and
$$ h(t)=h(t+T_2)  \quad  \Rightarrow  \sum_k \left(c_k \sin(2\pi k \frac{t}{T_2}) + d_k \cos (2\pi k \frac{t}{T_2}) \right) $$ 
If  $z(t)=g(t)+h(t)$
Under what circumstances of $T_1$ and $T_2$ would 
$$ z(t)=z(t+T_3) \quad \text{?}$$
an what circumstances of $T_1$ and $T_2$  would
$$
z(t)\ne z(t+T_3) \quad \text{for any} \quad T_3
$$

and if there were circumstance where $z(t)=z(t+T_3)$,

Could we re express $g(t)$ and $h(t)$ as Fourier Series periodic in $T_3$?