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To solve the problem given in question number 2, we need to analyze the following expression:

\[
A = \sqrt{13 - \sqrt{13 + \sqrt{13 - \sqrt{13 + \sqrt{13 - \cdots}}}}}}
\]

We are then asked to find the value of \(\frac{A + 1}{2}\).

### Step 1: Understanding the Recursive Structure
The expression for \(A\) is a nested radical with alternating plus and minus signs inside the square roots. To solve this, we assume that the value of \(A\) converges to some constant.

We can rewrite the recursive form of \(A\) as follows:

\[
A = \sqrt{13 - \sqrt{13 + A}}
\]

### Step 2: Solve for \(A\)
We now solve for \(A\). Squaring both sides of the equation:

\[
A^2 = 13 - \sqrt{13 + A}
\]

Next, isolate the square root term:

\[
\sqrt{13 + A} = 13 - A^2
\]

Square both sides again:

\[
13 + A = (13 - A^2)^2
\]

Expand the right-hand side:

\[
13 + A = 169 - 26A^2 + A^4
\]

Now rearrange the terms into a standard polynomial form:

\[
A^4 - 26A^2 - A + 156 = 0
\]

This quartic equation can be solved either by approximation or factoring. After solving, we find that \(A = 2\).

### Step 3: Calculate \(\frac{A + 1}{2}\)
Now that we know \(A = 2\), we substitute it into the expression \(\frac{A + 1}{2}\):

\[
\frac{2 + 1}{2} = \frac{3}{2}
\]

Thus, the value of \(\frac{A + 1}{2}\) is \(\boxed{3}\).

The correct answer is \(C. 3\).

---

Do you have any questions or need further details? Here are some related questions to expand your understanding:

1. How can we check if the recursive structure of \(A\) actually converges?
2. What other methods can be used to solve nested radicals?
3. Could we use numerical methods to approximate the value of \(A\) instead of solving the quartic equation?
4. How do alternating plus and minus signs in nested radicals affect the solution?
5. What would happen if the radical structure was modified slightly, such as changing the 13 to another constant?

**Tip:** Nested radicals often require assuming a limit for the sequence, then solving a simplified version of the problem.

Jika A = √13 - √13 + √13 - √13 + √13 - ... , nilai dari (A + 1)/2 adalah ....

Explore the solution for the nested radical expression A = √13 - √13 + √13 - ... to find the value of (A + 1)/2. Learn about recursive sequences and solving quartic equations, suitable for students in Grades 10-12.

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