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Chapter 1: Problem 109

Consider the sequence \(4,1 / 3,4 / 3,4 / 9,16 / 27,64 / 81, \ldots\), in which each term (after the first two) is the product of the two previous ones. Note that for this particular sequence, the first and third terms are greater than 1 while the second and fourth terms are less than 1 . However, after that the "alternating" pattern fails: the fifth and all subsequent terms are less than 1. Do there exist sequences of positive real numbers in which each term is the product of the two previous terms and for which all odd-numbered terms are greater than 1, while all even-numbered terms are less than 1? If so, find all such sequences. If not, prove that no such sequence is possible.

Short Answer

Expert verified
No, such sequence does not exist.

Step by step solution

01

Define the sequence terms

Let's define the sequence terms as follows: \(a_1 = 4\), \(a_2 = \frac{1}{3}\), \(a_3 = \frac{4}{3}\), and so on.
02

Understand the recursive relation

The sequence is defined by the relation \(a_{n+2} = a_{n+1} \times a_n\) for \(n \ge 1\). Each term is the product of the two previous terms.
03

Evaluate the pattern

Notice: For the given sequence, \(a_1 = 4\) (greater than 1), \(a_2 = \frac{1}{3}\) (less than 1), \(a_3 = \frac{4}{3}\) (greater than 1), and \(a_4 = \frac{4}{3} \times \frac{1}{3} = \frac{4}{9}\) (less than 1). The pattern alternates initially.
04

Examine subsequent terms

From the fifth term onward, observe \(a_5 = a_4 \times a_3 = \frac{4}{9} \times \frac{4}{3} = \frac{16}{27}\) (less than 1). Continued multiplication results in all terms remaining less than 1.
05

Determine if conditions can be met

To satisfy all odd-numbered terms being greater than 1 and all even-numbered terms being less than 1, note that the growth factor (alternating product) of terms must stabilize correctly.
06

Proof of Non-existence

Show that if \(a_1 > 1\) and \(a_2 < 1\), then for \(a_{n+2} = a_{n+1} \times a_n\), terms cannot sustain the even-odd conditions after initial terms due to continuous multiplication reducing value progressively.
07

Conclusion

Since the product of terms results in values being reduced consistently, no such sequence exists where all odd-numbered terms are greater than 1, while even-numbered terms are less than 1 indefinitely.

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Key Concepts

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

Recursive Sequences
A recursive sequence is a type of sequence where each term is defined based on the preceding terms. In the given problem, we see that each term after the first two is the product of the two previous terms. This is mathematically represented as \(a_{n+2} = a_{n+1} \times a_n\). With this recursive definition, to find any term in the sequence, we must know the two terms that come before it. Recursive sequences are often explored for their patterns and behaviors over long terms.
Real Numbers
Real numbers encompass all numbers that can be found on the number line, including both rational and irrational numbers. In this exercise, the terms of the sequence are all real numbers. More specifically, they are positive real numbers. Understanding whether a sequence of real numbers can maintain certain properties indefinitely is key to solving the problem. Here it is pertinent to understand both the broader characteristics of real numbers and the specific properties of sequences made up of such numbers.
Multiplicative Sequences
A multiplicative sequence is one where each term is generated by multiplying previous terms. As shown in the exercise, the sequence's recursive nature makes it multiplicative because each term is the product of the last two terms. For example, given the terms \(a_1 = 4\) and \(a_2 = \frac{1}{3}\), the third term is \(a_3 = a_2 \times a_1 = \frac{1}{3} \times 4 = \frac{4}{3}\). Each new term brings about changes in the values of subsequent terms, which is crucial in understanding the behavior of the sequence.
Alternating Patterns
An alternating pattern in sequences typically refers to a predictable change in behavior from one term to the next. In the problem, we initially observe an alternating pattern where odd-numbered terms are greater than 1 and even-numbered terms are less than 1. For instance, \(a_1 = 4 (greater than 1)\), \(a_2 = \frac{1}{3} (less than 1)\), \(a_3 = \frac{4}{3} (greater than 1)\), and so on. However, as the exercise progresses, it becomes clear that this pattern cannot be sustained indefinitely due to the nature of the multiplicative relationship, which gradually reduces all terms to less than 1.

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Most popular questions from this chapter

Every year, the first warm days of summer tempt Lake Wohascum's citizens to venture out into the local parks; in fact, one day last May, the MAA Student Chapter held an impromptu picnic. A few insects were out as well, and at one point an insect dropped from a tree onto a paper plate (fortunately an empty one) and crawled off. Although this did not rank with Newton's apple as a source of inspiration, it did lead the club to wonder: If an insect starts at a random point inside a circle of radius \(R\) and crawls in a straight line in a random direction until it reaches the edge of the circle, what will be the average distance it travels to the perimeter of the circle? ("Random point" means that given two equal areas within the circle, the insect is equally likely to start in one as in the other; "random direction" means that given two equal angles with vertex at the point, the insect is equally likely to crawl off inside one as the other.)

Suppose we have a configuration (set) of finitely many points in the plane which are not all on the same line. We call a point in the plane a center for the configuration if for every line through that point, there is an equal number of points of the configuration on either side of the line. a. Give a necessary and sufficient condition for a configuration of four points to have a center. b. Is it possible for a finite configuration of points (not all on the same line) to have more than one center?

As is well known, the unit circle has the property that distances along the curve are numerically equal to the difference of the corresponding angles (in radians) at the origin; in fact, this is how angles are often defined. (For example, a quarter of the unit circle has length \(\pi / 2\) and corresponds to an angle \(\pi / 2\) at the origin.) Are there other differentiable curves in the plane with this property? If so, what do they look like?

A particle starts somewhere in the plane and moves 1 unit in a straight line. Then it makes a "shallow right turn," abruptly changing direction by an acute angle \(\alpha\), and moves 1 unit in a straight line in the new direction. Then it again changes direction by \(\alpha\) (to the right) and moves 1 unit, and so forth. In all, the particle takes 9 steps of 1 unit each, with each direction at an angle \(\alpha\) to the previous direction. a. For which value(s) of \(\alpha\) does the particle end up exactly at its starting point? b. For how many values of the acute angle \(\alpha\) does the particle end up at a point whose (straight-line) distance to the starting point is exactly 1 unit?

Let \(g\) be a continuous function defined on the positive real numbers. Define a sequence \(\left(f_n\right)\) of functions as follows. Let \(f_0(x)=1\), and for \(n \geq 0\) and \(x>0\), let $$ f_{n+1}(x)=\int_1^x f_n(t) g(t) d t $$ Suppose that for all \(x>0, \sum_{n=0}^{\infty} f_n(x)=x\). Find the function \(g\). \(\quad\)
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