The image shows the following problem:

Problem:
Show that the sequence $\{x_n\}$ defined by
$x_1 = \sqrt{7}, \quad x_{n+1} = \sqrt{7 + x_n}$
converges to the positive root of the equation
$x^2 - x - 7 = 0.$

Step-by-Step Explanation:

1. Find the roots of the equation $x^2 - x - 7 = 0$:

   • The quadratic equation is given as $x^2 - x - 7 = 0$.
   • Using the quadratic formula:
   = \frac{1 \pm \sqrt{1 + 28}}{2} = \frac{1 \pm \sqrt{29}}{2}$$ - Therefore, the roots are: $$x = \frac{1 + \sqrt{29}}{2} \quad \text{(positive root)} \quad \text{and} \quad x = \frac{1 - \sqrt{29}}{2} \quad \text{(negative root)}.$$ Since the sequence is defined with square roots, we are interested in the **positive root**: $$x = \frac{1 + \sqrt{29}}{2}.$$

2. Show that the sequence converges:

   • We need to show that the sequence $\{x_n\}$, where $x_{n+1} = \sqrt{7 + x_n}$, converges to the positive root found above.

   Let's assume the sequence converges to a limit $L$. If the sequence converges, then in the limit as $n \to \infty$, we have: $L = \sqrt{7 + L}.$ Squaring both sides: $L^2 = 7 + L.$ Rearranging the terms: $L^2 - L - 7 = 0.$ This is exactly the equation $x^2 - x - 7 = 0$, whose positive root we found to be $L = \frac{1 + \sqrt{29}}{2}$.

3. Conclusion:

   • The sequence $\{x_n\}$, defined by $x_1 = \sqrt{7}$ and $x_{n+1} = \sqrt{7 + x_n}$, converges to the positive root of the equation $x^2 - x - 7 = 0$, which is $\frac{1 + \sqrt{29}}{2}$.

Would you like to see further details or ask any other questions?

Here are five related questions you might want to explore:

1. What are the conditions for convergence of a recursive sequence?
2. How does the initial value $x_1 = \sqrt{7}$ affect the convergence?
3. Can this method be generalized to other quadratic equations?
4. What is the behavior of the negative root of the equation in the context of this sequence?
5. What are other examples of sequences that converge to a root of an equation?

Tip: To ensure convergence in sequences like this one, check if the recursive relation is a contraction, meaning it brings values closer together as $n$ increases.