Given the recurrence relation Un+2 = 5Un+1 − 6Un with u1 = 1 and u2 = 2, find the value of u4.

Set $n=1$ in the recurrence relation:

$$U_{1+2} = 5U_{1+1} - 6U_1$$ $$U_3 = 5U_2 - 6U_1$$

Substitute the given values $u_1 = 1$ and $u_2 = 2$:

$$U_3 = 5(2) - 6(1)$$ $$U_3 = 10 - 6$$ $$U_3 = 4$$

Set $n=2$ in the recurrence relation:

$$U_{2+2} = 5U_{2+1} - 6U_2$$ $$U_4 = 5U_3 - 6U_2$$

Substitute the calculated value $u_3 = 4$ and the given value $u_2 = 2$:

$$U_4 = 5(4) - 6(2)$$ $$U_4 = 20 - 12$$ $$U_4 = 8$$

The value of $u_4$ is $\boxed{8}$.