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.