For the sequence defined by Un+1 = 1 + 1/2 Un, find a formula for Un using the general method, expressing it…

Question

For the sequence defined by Un+1 = 1 + 1/2 Un, find a formula for Un using the general method, expressing it in terms of U1.

ScanSolve AI Tutor · Accepted Answer

Here's how to solve parts (b) and (c) for the sequence defined by $U_{n+1} = 1 + \frac{1}{2} U_n$ . We will express the formulas in terms of the initial term $U_1$ , as it is not specified for this new recurrence relation.

b) Find a formula for $U_n$ using the general method.

Find the homogeneous solution.
The homogeneous part of the recurrence relation is $U_{n+1} = \frac{1}{2} U_n$ .
The characteristic equation is $r = \frac{1}{2}$ .
Thus, the homogeneous solution is $U_n^{(h)} = A \left(\frac{1}{2}\right)^n$ , where $A$ is a constant.
Find the particular solution.
Assume a constant particular solution $U_n^{(p)} = C$ . Substitute this into the original recurrence relation:
$C = 1 + \frac{1}{2} C$ $C - \frac{1}{2} C = 1$ $\frac{1}{2} C = 1$ $C = 2$
So, the particular solution is $U_n^{(p)} = 2$ .
Combine to find the general solution.
The general solution is the sum of the homogeneous and particular solutions:
$U_n = U_n^{(h)} + U_n^{(p)}$ $U_n = A \left(\frac{1}{2}\right)^n + 2$
Determine the constant $A$ using the initial term $U_1$ .
For $n=1$ :
$U_1 = A \left(\frac{1}{2}\right)^1 + 2$ $U_1 = \frac{A}{2} + 2$ $\frac{A}{2} = U_1 - 2$ $A = 2(U_1 - 2)$
Substitute $A$ back into the general

For the sequence defined by Un+1 = 1 + (1)/(2) Un, find a formula for Un using the general method, expressing it in terms of U1.

Step-by-step solution

Find the homogeneous solution.

Find the particular solution.

Combine to find the general solution.

Determine the constant $A$ using the initial term $U_1$ .

Substitute $A$ back into the general

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