It seems like you're working on a recursive function problem with an initial condition and a function definition. The problem provides a function $f(x) = \frac{1}{11}x + 20$, and you're asked to compute values for $x_1, x_2, x_3, x_4, x_5$, with the recursive relation $x_n = f(x_{n-1})$ starting with $x_0 = 0$.

Let's go ahead and calculate each of the values:

1. Start with $x_0 = 0$.
2. Compute $x_1 = f(x_0)$.
3. Use the result to compute $x_2 = f(x_1)$, and so on until $x_5$.

I'll go ahead and compute the values now. Function definition def f(x): return (1 / 11) * x + 20

Initial value

x_0 = 0

Compute x1 to x5 using the recursive relation

x_values = [x_0] for n in range(1, 6): x_values.append(f(x_values[-1]))

Round the results to five decimal places

x_values_rounded = [round(x, 5) for x in x_values] x_values_roundedThe computed values for $x_1, x_2, x_3, x_4, x_5$ are:

• $x_1 = 20.00000$
• $x_2 = 21.81818$
• $x_3 = 21.98347$
• $x_4 = 21.99850$
• $x_5 = 21.99986$

As you can observe, the values are approaching 22 as $n$ increases.

Conjecture:

As $n \to \infty$, the value of $x_n$ approaches 22.

Would you like a deeper explanation of the process or further assistance with this problem? Here are some related questions you might find interesting:

1. What happens if we change the function $f(x)$?
2. How would the result change if the initial value $x_0$ were different?
3. Can this method be used for other types of recursive sequences?
4. What is the significance of the limiting value 22 in this context?
5. How does the function $f(x)$ affect the convergence of the sequence?

Tip: For recursive functions like this, you can often analyze the behavior of the sequence by identifying fixed points where $f(x) = x$.