Question

Consider the following recurrence relations. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. $$a_{n+1}=\frac{a_{n}}{2} ; a_{0}=32$$

Short answer

Answer: The limit of the sequence is 0.

Step-by-step solution

Calculate the first 10 terms of the sequence

To calculate each term of the sequence, use the given recurrence relation: $$a_{n+1}=\frac{a_{n}}{2}$$ 1. \(a_1 = \frac{a_0}{2} = \frac{32}{2} = 16\) 2. \(a_2 = \frac{a_1}{2} = \frac{16}{2} = 8\) 3. \(a_3 = \frac{a_2}{2} = \frac{8}{2} = 4\) 4. \(a_4 = \frac{a_3}{2} = \frac{4}{2} = 2\) 5. \(a_5 = \frac{a_4}{2} = \frac{2}{2} = 1\) 6. \(a_6 = \frac{a_5}{2} = \frac{1}{2} = 0.5\) 7. \(a_7 = \frac{a_6}{2} = \frac{0.5}{2} = 0.25\) 8. \(a_8 = \frac{a_7}{2} = \frac{0.25}{2} = 0.125\) 9. \(a_9 = \frac{a_8}{2} = \frac{0.125}{2} = 0.0625\) 10. \(a_{10} = \frac{a_9}{2} = \frac{0.0625}{2} = 0.03125\)

Create a table with calculated terms

Create a table of the calculated terms: | Term (\(n\)) | Value (\(a_n\)) | |----|----| | 0 | 32 | | 1 | 16 | | 2 | 8 | | 3 | 4 | | 4 | 2 | | 5 | 1 | | 6 | 0.5| | 7 | 0.25| | 8 | 0.125| | 9 | 0.0625| | 10 | 0.03125|

Determine the limit of the sequence

Looking at the table, it appears that the values of \(a_n\) are getting closer and closer to 0 as \(n\) increases. We can see that each term is half of the previous one, so the sequence will keep getting smaller and smaller but never reach 0. Therefore, a plausible value for the limit of the sequence is 0.

Key concepts

Sequence Limit

In mathematics, the concept of a "sequence limit" is fundamental when dealing with patterns or recurrences. A sequence limit is essentially the value that the terms in a sequence approach as the number of terms increases indefinitely. Consider our example sequence, where each term is half the previous one, starting from 32: 16, 8, 4, and so on. As we continue this pattern, the terms get closer and closer to 0, even though 0 is never actually reached. Hence, we say the limit of this sequence is 0.

Understanding sequence limits helps us identify the behavior of sequences as they progress. It is crucial in various fields such as calculus and real analysis, providing insight into the long-term behavior of infinite lists of numbers.

• Sequence limit allows prediction of pattern behavior over time.
• It's especially useful in real-world applications like physics and engineering.
• Identifying limits helps in assessing the convergence or divergence of a sequence.

Sequence Convergence

A critical concept related to sequence limits is called "sequence convergence." A sequence is said to converge if it approaches a specific limit as the number of its terms goes to infinity. In our recurrence relation example, where we started with 32 and continued generating terms by halving the previous term, the sequence converges to a limit of 0.

Convergence is a property that allows mathematicians to understand stability and predictability in mathematical models. In practical terms, determining if a sequence converges can indicate whether it settles into a pattern or keeps changing unpredictably.

• Convergence indicates stability in sequences, which is key in mathematical modeling.
• Knowing the point of convergence allows us to understand long-term implications of a sequence's behavior.
• Some sequences, unlike our example, may not converge, indicating more complex behaviors.

Calculus

When discussing sequence limits and convergence, we often venture into the realm of calculus. Calculus is the branch of mathematics that studies continuous change and includes the study of limits, derivatives, and integrals. It provides tools for understanding complex systems and changes, making it applicable across various sciences.

The concept of a limit, used in analyzing sequences, is foundational in calculus. Calculus not only helps define the limit of a sequence but also aids in understanding the rate of change within that sequence.

• Calculus helps connect discrete math with continuous processes by analyzing limits.
• It provides techniques for computing real-world solutions involving areas, volumes, and rates of change.
• Understanding the fundamentals of calculus empowers one with skills to tackle extremely diverse and real-world complex problems.