To verify the given results from the image, let us carefully analyze the provided mathematical problem and the accompanying work:

Problem Overview:

The task involves analyzing the convergence of a recursively defined sequence $a_n$ given by: $a_1 = 3, \quad a_{n+1} = \sqrt{7a_n + 8}, \quad \text{for all } n \in \mathbb{N}.$

You are required to:

1. Determine the candidate for the potential limit of the sequence.
2. Prove boundedness and monotonicity, and ultimately establish whether the sequence converges.

Step-by-Step Review

1. Candidate for the Limit ($a$)

   • The potential limit $a$ satisfies: $a = \sqrt{7a + 8}.$ Squaring both sides: $a^2 = 7a + 8 \implies a^2 - 7a - 8 = 0.$ Factoring or solving the quadratic equation gives: $a = \frac{7 \pm \sqrt{49 + 32}}{2} = \frac{7 \pm 9}{2}.$ Hence, the solutions are: $a_1 = 8 \quad \text{and} \quad a_2 = -1.$ Since the sequence $a_n$ is positive, the only viable candidate is: $a = 8.$

   ✅ The candidate for the limit $a$ is correctly identified as $8$.

2. Boundedness

   • Initial analysis shows $a_n > 0$ for all $n$, so the sequence is bounded below by $0$.

   • To prove the sequence is bounded above by $b = 8$, mathematical induction is used:

     Inductive Step: Assume $a_n \leq 8$. Then: $a_{n+1} = \sqrt{7a_n + 8} \leq \sqrt{7 \cdot 8 + 8} = \sqrt{64} = 8.$ Hence, $a_{n+1} \leq 8$, proving the sequence is bounded above.

   ✅ The boundedness proof appears correct.

3. Monotonicity

   • To check if the sequence is monotonic, evaluate whether $a_{n+1} \geq a_n$ (monotonically increasing): $a_{n+1} - a_n = \sqrt{7a_n + 8} - a_n.$ For $a_n < 8$, note that $\sqrt{7a_n + 8} > a_n$, which confirms the sequence is increasing.

   ✅ The sequence is correctly identified as monotonically increasing.

4. Convergence

   • Since the sequence is bounded and monotonically increasing, it converges by the Monotone Convergence Theorem. The limit is the candidate $a = 8$, as verified earlier.

   ✅ The convergence and limit are correctly determined.

If you would like, I can provide additional clarification or rewrite the solution with detailed explanations.

Related Questions

1. What other methods can you use to check boundedness?
2. How do you prove monotonicity in recursive sequences in general?
3. Can this approach be generalized for other quadratic recurrence relations?
4. What is the role of the Monotone Convergence Theorem in such problems?
5. How would the results differ if the initial condition $a_1$ were negative?

Tip: When solving recursive sequences, always verify initial terms to ensure the sequence remains within the derived bounds.