35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 
{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50;a₀ = 50

Verified step by step guidance

1. Identify the given recursive sequence: \(a_{n+1} = \frac{a_n}{11} + 50\) with initial term \(a_0 = 50\).
2. Calculate the first four terms by substituting the previous term into the recursive formula: - \(a_1 = \frac{a_0}{11} + 50\) - \(a_2 = \frac{a_1}{11} + 50\) - \(a_3 = \frac{a_2}{11} + 50\) - \(a_4 = \frac{a_3}{11} + 50\).
3. Write each term explicitly by performing the substitution step-by-step, but do not simplify the numerical values yet.
4. To analyze convergence, assume the sequence converges to a limit \(L\). Then, by the definition of limit for recursive sequences, set \(L = \frac{L}{11} + 50\).
5. Solve the equation \(L = \frac{L}{11} + 50\) for \(L\) to find the conjectured limit of the sequence. This will help determine if the sequence converges or diverges.

Key Concepts

Sequences and Their Terms

A sequence is an ordered list of numbers defined by a specific rule. Each term, denoted as aₙ, depends on its position n. Understanding how to compute initial terms using the given recursive formula is essential to analyze the sequence's behavior.

Limits and Convergence of Sequences

A sequence converges if its terms approach a specific finite value as n becomes very large. The limit is this value. Recognizing whether a sequence converges or diverges helps in predicting long-term behavior and making conjectures about the limit.

Recursive Sequences and Fixed Points

Recursive sequences define each term based on previous terms. A fixed point is a value that remains unchanged when plugged into the recursive formula. Finding fixed points helps determine possible limits of the sequence if it converges.