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The given functional equation is \( f(x) = f(x-1) + f(x+1) \), with initial conditions \( f(6) = 6 \) and \( f(10) = 3f(11) \). We are tasked with finding \( f(20) \).

### Step 1: Simplify the given functional equation
We have:
\[
f(x) = f(x-1) + f(x+1).
\]
Rearranging this, we obtain:
\[
f(x+1) = f(x) - f(x-1).
\]
This equation indicates a recursive relation between successive values of \( f \).

### Step 2: Using the conditions
We are given the conditions:
\[
f(6) = 6 \quad \text{and} \quad f(10) = 3f(11).
\]
Let's use these to explore the values of the function:

#### Condition 1: \( f(6) = 6 \)
This tells us the value of \( f \) at \( x = 6 \).

#### Condition 2: \( f(10) = 3f(11) \)
We'll use this later once we find a pattern in the function.

### Step 3: Calculate successive values of \( f(x) \)
Let's calculate some values of \( f(x) \) using the recurrence relation:

- Start with \( f(6) = 6 \).
- Using the functional equation, we calculate the nearby values of \( f \):
  - \( f(7) = f(6) - f(5) \).
  - \( f(8) = f(7) - f(6) \), and so on.

Let's continue this calculation step-by-step.It seems I neglected to calculate the values of \( f(5) \) and earlier ones. Let's fix that by computing values step-by-step, starting from the known \( f(6) = 6 \). I'll extend the recursion both forward and backward to find \( f(20) \) and use the condition \( f(10) = 3f(11) \).It seems we also need to calculate values further backward from \( f(5) \). I'll adjust the approach and properly handle the recursive relation to compute all necessary values step by step from \( f(6) \), ensuring the logic follows correctly for both backward and forward calculations. Let me handle that now.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to try working through the problem manually, or perhaps we can try a different approach? Let me know if you'd like assistance with any other math questions or details about this specific problem!

Here are some related questions to explore:
1. How can you solve a recursive functional equation step-by-step?
2. What does it mean when a function satisfies a recurrence relation?
3. Can initial conditions fully determine a function in recurrence relations?
4. How does the condition \( f(10) = 3f(11) \) help in solving this problem?
5. What methods can be used to simplify recursive relations for large values?

**Tip:** In functional equations, finding patterns for small values of \( x \) can reveal a general rule that simplifies the solution for larger values.

If f(x) = f(x-1) + f(x+1), where f(6) = 6 and f(10) = 3f(11), find f(20).

Learn how to solve the recursive functional equation f(x) = f(x-1) + f(x+1) with given conditions f(6) = 6 and f(10) = 3f(11) to find f(20). This problem explores recursion and functional equations suitable for high school students in Grades 11-12.

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