Les 2 exercices et le problème sont obligatoires - Machine à calculer scientifique non programmable autorisée.

For a geometric sequence, the term $U_n$ can be expressed as $U_n = U_p \cdot r^{n-p}$.
Given $U_3 = 2$ and $U_6 = \frac{1}{4}$, we can write:

$$U_6 = U_3 \cdot r^{6-3}$$ $$\frac{1}{4} = 2 \cdot r^3$$

Divide both sides by 2:

$$r^3 = \frac{1}{4 \times 2}$$ $$r^3 = \frac{1}{8}$$

Take the cube root of both sides:

$$r = \sqrt[3]{\frac{1}{8}}$$ $$r = \frac{1}{2}$$

We use the formula $U_n = U_0 \cdot r^n$. Using $U_3 = 2$ and $r = \frac{1}{2}$:

$$U_3 = U_0 \cdot r^3$$ $$2 = U_0 \cdot \left(\frac{1}{2}\right)^3$$ $$2 = U_0 \cdot \frac{1}{8}$$

Multiply both sides by 8:

$$U_0 = 2 \times 8$$ $$U_0 = 16$$

The common ratio is $\boxed{r = \frac{1}{2}}$ and the first term is $\boxed{U_0 = 16}$.