The problem asks us to use the iteration formula:

$x_{n+1} = \sqrt{13 + 5x_n}$

with the initial value $x_0 = 6$ to find $x_3$ to 5 decimal places.

Step-by-step solution:

1. Initial value: $x_0 = 6$

2. Calculate $x_1$: Using the iteration formula: $x_1 = \sqrt{13 + 5x_0} = \sqrt{13 + 5(6)} = \sqrt{13 + 30} = \sqrt{43}$ Calculating $\sqrt{43}$: $x_1 \approx 6.55743852$

3. Calculate $x_2$: Using $x_1$ in the iteration formula: $x_2 = \sqrt{13 + 5x_1} = \sqrt{13 + 5(6.55743852)} = \sqrt{13 + 32.7871926} = \sqrt{45.7871926}$ Calculating $\sqrt{45.7871926}$: $x_2 \approx 6.769647$

4. Calculate $x_3$: Using $x_2$ in the iteration formula: $x_3 = \sqrt{13 + 5x_2} = \sqrt{13 + 5(6.769647)} = \sqrt{13 + 33.848235} = \sqrt{46.848235}$ Calculating $\sqrt{46.848235}$: $x_3 \approx 6.843159$

Thus, $x_3 \approx 6.84316$ to 5 decimal places.

Would you like more details on any step? Here are 5 related questions you might consider:

1. How would the convergence of the sequence $x_n$ be affected by a different starting point $x_0$?
2. Can this iterative method be used to find roots of equations? If yes, how?
3. What is the importance of precision in iterative methods like this one?
4. How would the iteration formula change if we used different constants in the equation?
5. How could we determine when to stop iterating if no decimal precision was specified?

Tip: Iterative methods often converge faster if the starting value is closer to the true solution.