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Chapter 8: Problem 39

Write the terms \(a_{1}, a_{2}, a_{3},\) and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. $$a_{n+1}=\frac{a_{n}}{11}+50 ; a_{0}=50$$

Short Answer

Expert verified
If it converges, what is the limit? Answer: The sequence appears to converge, and the conjectured limit is 55.

Step by step solution

01

Find the first term, \(a_{1}\)

To find the first term \(a_{1}\), we will use the given formula and the initial term, \(a_{0}\): $$a_{1}=\frac{a_{0}}{11}+50=\frac{50}{11}+50 \approx 54.54$$
02

Find the second term, \(a_{2}\)

Using the formula and \(a_{1}\), we find the second term \(a_{2}\): $$a_{2}=\frac{a_{1}}{11}+50=\frac{54.54}{11}+50 \approx 54.95$$
03

Find the third term, \(a_{3}\)

Similarly, using the formula and \(a_{2}\), we find the third term \(a_{3}\): $$a_{3}=\frac{a_{2}}{11}+50=\frac{54.95}{11}+50 \approx 54.99$$
04

Find the fourth term, \(a_{4}\)

Finally, using the formula and \(a_{3}\), we find the fourth term \(a_{4}\): $$a_{4}=\frac{a_{3}}{11}+50=\frac{54.99}{11}+50 \approx 55$$
05

Analyze the sequence for convergence or divergence

Looking at the terms \(a_{1}, a_{2}, a_{3},\) and \(a_{4}\), we observe that they are approximately 54.54, 54.95, 54.99, and 55, respectively. The terms seem to be increasing gradually but at a decreasing rate, thus suggesting that the sequence might be converging to a certain limit. Furthermore, the terms are approaching 55.
06

Conclusion

The first four terms of the sequence are approximately 54.54, 54.95, 54.99, and 55. Based on these terms, the sequence appears to converge, and we conjecture that the limit is 55.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence of Sequences
In mathematics, understanding whether a sequence is converging is key to many problems. A sequence is said to be convergent if its terms approach a specific value as they progress. This ultimate value, if reached, is referred to as the sequence's limit. Convergence is a fundamental concept because it helps us predict the behavior of sequences over time.
For the given exercise, the sequence \( a_{n+1} = \frac{a_n}{11} + 50 \) shows terms that steadily move closer together. They do this at a decreasing rate and appear to be getting nearer to 55. This pattern indicates convergence because:
  • The terms do not wildly fluctuate, but instead continuously move towards the same value.

  • The difference between consecutive terms tends to shrink over time.
Convergence is a powerful tool for validating assumptions about mathematical models, as it allows for predictions about long-term behavior.
Limit of a Sequence
The limit of a sequence is the value that the terms of the sequence approach as the index becomes very large. In simple terms, it is the number the sequence is trying to reach. When a sequence converges to a limit, every term gets closer than ever to this number but might never actually reach it.
A limit acts as a way to describe the end behavior of a sequence and establishes whether a sequence stabilizes. In our sequence \( a_{n+1} = \frac{a_n}{11} + 50 \), as the terms \( a_1, a_2, a_3, \) and beyond become closer to 55, it's evident that the sequence has a limit at this number. The key points to identify a limit are:
  • Look for a pattern in successive terms - they should seem to stabilize.

  • The terms should hover around a number with a minimal change.
Recognizing limits helps forecast sequences in various fields, including science and engineering.
Recurrence Relations
Recurrence relations are equations that define sequences based on previous terms. They serve as rules that dictate how a sequence develops step by step. In our problem, the given recurrence relation is \( a_{n+1} = \frac{a_n}{11} + 50 \), with \( a_0 = 50 \). This means each term is computed from its predecessor.
These relations are essential for generating predictive models, such as in finance, computer science, and natural phenomena. There are some important aspects to remember about recurrence relations:
  • Identify the initial condition(s) - here it's \( a_0 = 50 \).

  • Understand the pattern or rule used to get to the next term from the current one.
With recurrence relations, we gain a structured way to step through sequences, thereby simplifying complex problems through iterative computation.

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Most popular questions from this chapter

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{3^{n}}{n !}\right\\}$$

Imagine that the government of a small community decides to give a total of \(\$ W\), distributed equally, to all its citizens. Suppose that each month each citizen saves a fraction \(p\) of his or her new wealth and spends the remaining \(1-p\) in the community. Assume no money leaves or enters the community, and all the spent money is redistributed throughout the community. a. If this cycle of saving and spending continues for many months, how much money is ultimately spent? Specifically, by what factor is the initial investment of \(\$ W\) increased (in terms of \(p\) )? Economists refer to this increase in the investment as the multiplier effect. b. Evaluate the limits \(p \rightarrow 0\) and \(p \rightarrow 1,\) and interpret their meanings.

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=n^{10} \text { and } b_{n}=n^{9} \ln ^{3} n, n \geq 7$$

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty}(-1)^{k}$$

The Greeks solved several calculus problems almost 2000 years before the discovery of calculus. One example is Archimedes' calculation of the area of the region \(R\) bounded by a segment of a parabola, which he did using the "method of exhaustion." As shown in the figure, the idea was to fill \(R\) with an infinite sequence of triangles. Archimedes began with an isosceles triangle inscribed in the parabola, with area \(A_{1}\), and proceeded in stages, with the number of new triangles doubling at each stage. He was able to show (the key to the solution) that at each stage, the area of a new triangle is \(\frac{1}{8}\) of the area of a triangle at the previous stage; for example, \(A_{2}=\frac{1}{8} A_{1},\) and so forth. Show, as Archimedes did, that the area of \(R\) is \(\frac{4}{3}\) times the area of \(A_{1}\).
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