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Chapter 9: Problem 16

Consider the sequence \(3,2, \frac{5}{4}, \frac{6}{8}, \frac{7}{16}, \ldots\) (a) List the next two terms of the sequence. (b) If the notation for the sequence is \(A_{1}, A_{2}, A_{3}, \ldots,\) give an explicit formula for \(A_{N}\)

Short Answer

Expert verified
The next two terms of the sequence are \(\frac{1}{4}\) and \(\frac{9}{64}\). The explicit formula for any term \(A_N\) of the sequence \(A_1, A_2, A_3, \ldots\) is \(A_N = \frac{N+2}{2^{N-1}}\).

Step by step solution

01

Identify the pattern and list the next two terms

After recognizing the pattern, the next two terms can be calculated. The sixth term is \(\frac{7}{16}\) which is obtained by dividing 7 (which is \(6+1\)) by \(2^4\). Thus, the next term is \(\frac{8}{32} = \frac{1}{4}\) and the term after that is \(\frac{9}{64}\).
02

Formulate an explicit formula for the sequence

An explicit formula can be derived from the sequence. Note that the first term \(A_1\) is 3 which could be written as \(\frac{4}{2^0}\), the second term \(A_2\) is 2 which is \(\frac{3}{2^1}\), the third term \(A_3\) is \(\frac{5}{4}\) which is \(\frac{3+2}{2^2}\) and so on. It becomes clear that the \(N^{th}\) term of the sequence can be given by \(A_N = \frac{N+2}{2^{N-1}}\) for all \(N \geq 1\). This formula captures both cases for \(N=1,2\) and \(N鈮�3\).

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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 sequence of numbers where each term is defined in relation to one or more previous terms. Instead of providing a direct expression for any given term, we often start with some initial terms and a rule that explains how to get the successive terms. Identifying a recursive rule requires observing the changes from term to term.
For our sequence, each term seems to be constructed by adding 1 to the numerator of the previous term and dividing by a power of 2 that increases steadily. This means that each term relies on its predecessor, building a chain of values that forms the sequence. Recursive sequences like this are common in mathematical problems and can help in formulating explicit expressions, which we did further in the solution.
Mathematical Formulas
Mathematical formulas are used to express calculations or relationships in a concise form. In the context of sequences, an explicit formula helps us find any term without having to compute all previous terms. It serves as a powerful tool in predicting and understanding complex patterns.
In our example sequence, the goal was to find an explicit formula for the sequence given in the problem: \( A_N = \frac{N+2}{2^{N-1}} \). This formula was derived by identifying how the numerators increase by 1 for each successive term and how the denominator follows a power of 2 pattern. Such explicit formulas simplify calculations and illustrate the beauty and efficiency of mathematical expressions.
Pattern Recognition
Pattern recognition is the process of identifying trends or regularities within data. This skill is crucial when dealing with sequences, as it allows us to find underlying rules that govern the formation of terms. Recognizing patterns can lead to the discovery of both recursive rules and explicit formulas.
In our provided sequence, careful observation revealed a steady increase of the numerator by 1 each time, and a consistent doubling of the denominator. By recognizing these patterns, we were able to predict future terms and derive an explicit formula. Practice in identifying such patterns sharpens problem-solving skills and enhances mathematical understanding.

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Most popular questions from this chapter

A population decays according to an exponential growth model, with \(P_{0}=3072\) and common ratio \(R=0.75 .\) (a) Find \(P_{5}\). (b) Give an explicit formula for \(P_{N}\). (c) How many generations will it take for the population

Consider the sequence defined by the explicit formula \(A_{N}=\frac{4 N}{N+3}\) (a) Find \(A_{1}\). (b) Find \(A_{9}\). (c) Suppose \(A_{N}=\frac{5}{2},\) Find \(N\).

18\. When two fair coins are tossed the probability of tossing two heads is \(\frac{1}{4}\), when three fair coins are tossed the probability of tossing two heads and one tail is \(\frac{3}{8} ;\) when four fair coins are tossed the probability of tossing two heads and two tails is \(\frac{6}{16}\) when five fair coins are tossed the probability of tossing two heads and three tails is \(\frac{10}{32}\). Using the sequence \(\frac{1}{4}, \frac{3}{8}, \frac{6}{16}, \frac{10}{32}, \ldots\) as your guide. (a) determine the probability of tossing two heads and four tails when six fair coins are tossed. (b) determine the probability of tossing two heads and 10 tails when 12 fair coins are tossed. (Hint: Find an explicit formula first.)

While the number of smokers for the general adult population is decreasing, it is not decreasing equally across all subpopulations (and for some groups it is actually increasing). For the 18 -to- 24 age group, the number of smokers was 8 million in 1965 and 6.3 million in \(2009 .\) (Source: "Trends in Tobacco Use," American Lung Association, 2011.) Assuming the number of smokers in the \(18-\) to- 24 age group continues decreasing according to a negative linear growth model, (a) predict the number of smokers in the \(18-\) to -24 age group in 2015 (round your answer to the nearest thousand). (b) predict in what year the number of smokers in the 18 to- 24 age group will reach 5 million.

A manufacturer currently has on hand 387 widgets. During the next 2 years, the manufacturer will be increasing his inventory by 37 widgets per week. (Assume that there are exactly 52 weeks in one year.) Each widget costs 10 cents a week to store. (a) How many widgets will the manufacturer have on hand after 20 weeks? (b) How many widgets will the manufacturer have on hand after \(N\) weeks? (Assume \(N \leq 104\).) (c) What is the cost of storing the original 387 widgets for 2 years (104 weeks)? (d) What is the additional cost of storing the increased inventory of widgets for the next 2 years?
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