To solve the given problem, let’s work through the iteration step by step:

The formula is: $x_{n+1} = 8 + \frac{500}{x_n^2}$ We are given $x_1 = 59$ and need to compute $x_3$, rounding each value to 2 decimal places.

Step 1: Calculate $x_2$

Substitute $x_1 = 59$ into the formula: $x_2 = 8 + \frac{500}{59^2}$ First, calculate $59^2$: $59^2 = 3481$ Now compute $\frac{500}{3481}$: $\frac{500}{3481} \approx 0.1436$ Add 8 to this result: $x_2 = 8 + 0.1436 = 8.1436$ Round to 2 decimal places: $x_2 \approx 8.14$

Step 2: Calculate $x_3$

Substitute $x_2 = 8.14$ into the formula: $x_3 = 8 + \frac{500}{(8.14)^2}$ First, calculate $(8.14)^2$: $(8.14)^2 \approx 66.2596$ Now compute $\frac{500}{66.2596}$: $\frac{500}{66.2596} \approx 7.546$ Add 8 to this result: $x_3 = 8 + 7.546 = 15.546$ Round to 2 decimal places: $x_3 \approx 15.55$

Final Answer:

The value of $x_3$ is: $\boxed{15.55}$

Let me know if you’d like further explanation or verification of any step!

Related Questions:

1. How would you determine $x_4$ using the same formula?
2. What happens to the sequence as $n$ increases (convergence or divergence)?
3. If the formula changes slightly, such as $x_{n+1} = 8 + \frac{k}{x_n^2}$, how does the value of $k$ impact the result?
4. What is the purpose of iterative formulas like this in mathematics?
5. How can you write this iterative formula in terms of a programming loop for computation?

Tip:

When working with iterative formulas, always carry out intermediate steps to a few extra decimal places before rounding to avoid rounding errors accumulating.