Find a general formula for Un for the sequence defined by Un+1 = 3 + rac14 Un with U1 = 5.

The homogeneous part of the recurrence relation is $U_{n+1} = \frac{1}{4} U_n$.
The characteristic equation is $r = \frac{1}{4}$.
The homogeneous solution is $U_n^{(h)} = A \left(\frac{1}{4}\right)^n$, where $A$ is a constant.

Assume a constant particular solution $U_n^{(p)} = C$. Substitute this into the original recurrence relation:

Subtract $\frac{1}{4} C$ from both sides:

Multiply by $\frac{4}{3}$:

So, the particular solution is $U_n^{(p)} = 4$.

The general solution is the sum of the homogeneous and particular solutions:

Substitute $n=1$ and $U_1 = 5$ into the general solution:

Subtract 4 from both sides:

Multiply by 4:

This can be simplified:

The formula for $U_n$ is $\boxed{U_n = \frac{1}{4^{n-1}} + 4}$.