Textbook Question
21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 2aₙ; a₁ = 2
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14. Sequences & Series
Sequences
2:10 minutesProblem 10.1.43
Textbook Question
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 guidance1
Identify the given recursive sequence: \(a_{n+1} = \frac{a_n}{11} + 50\) with initial term \(a_0 = 50\).
Calculate the first four terms by substituting the previous term into the recursive formula:
- \(a_1 = \frac{a_0}{11} + 50\)
- \(a_2 = \frac{a_1}{11} + 50\)
- \(a_3 = \frac{a_2}{11} + 50\)
- \(a_4 = \frac{a_3}{11} + 50\).
Write each term explicitly by performing the substitution step-by-step, but do not simplify the numerical values yet.
To analyze convergence, assume the sequence converges to a limit \(L\). Then, by the definition of limit for recursive sequences, set \(L = \frac{L}{11} + 50\).
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.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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