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Chapter 10: Problem 92

Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences in Exercises \(89-92 .\) What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{3 n^{4}+n-1}{5 n^{4}+2 n^{2}+1} \quad n:[0,10,1] \text { by } a_{n}:[0,1,0.1]$$

Short Answer

Expert verified
The graph might appear to approach a certain value or limit as \(n\) grows larger. By mathematically analyzing the sequence, it could be observed whether the sequence increases or decreases, or oscillates, or tends towards a certain value or limit. One can also deduce from the function that given a high enough \(n\), \(a_{n}\) essentially becomes \(\frac{3}{5}\) as higher powers of \(n\) in denominator and numerator will mainly influence the value.

Step by step solution

01

Understanding the Sequence

The sequence given is \(a_{n}\), whose terms are determined by the function \(\frac{3 n^{4}+n-1}{5 n^{4}+2 n^{2}+1}\). The question asks to observe and analyze what happens when \(n\) grows larger i.e. the behavior of the sequence.
02

Plotting the Sequence

As per the problem, \(n\) belongs to range [0,10,1] and \(a_{n}\) belongs to range [0,1,0.1], which means \(n\) takes values between 0 and 10 in steps of 1 and \(a_{n}\) takes values between 0 to 1 in steps of 0.1. Using a graphing tool, plot the sequence by mapping each value of \(n\) to the corresponding \(a_{n}\), which is obtained by substituting each \(n\) into the function.
03

Analyzing the Graph

Once the graph is plotted, analyze the pattern of the sequence. Check if the sequence is increasing or decreasing as \(n\) gets larger, or if it is oscillating between certain values, or if it is approaching a certain value, also known as a limit.

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Key Concepts

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

Graphing Utilities
Graphing utilities are powerful tools that help visualize various mathematical concepts, such as sequences, by plotting them as points on a graph. These tools are often part of graphing calculators or software programs. By inputting a sequence formula, the graphing utility can automatically calculate and display the corresponding terms on a graph. This visual representation is essential for understanding complex patterns and behaviors of sequences.

Using graphing utilities for sequences involves a few simple steps:
  • Input the sequence formula, in this case, \( a_{n} = \frac{3n^{4} + n - 1}{5n^{4} + 2n^{2} + 1} \).
  • Set the range for \( n \) and the desired increments, such as from 0 to 10 in steps of 1.
  • Adjust the graph viewing window to appropriately display the values of the sequence, ensuring clarity.
Graphing utilities allow users to visually interpret the behavior of sequences efficiently and accurately, aiding in deeper comprehension of mathematical patterns.
Rectangular Coordinate System
The rectangular coordinate system is a two-dimensional plane where points are defined by horizontal (x-axis) and vertical (y-axis) coordinates. This system is commonly used for graphing functions and sequences. Each point in this system can be represented as an ordered pair \((x, y)\), where \(x\) is the position along the horizontal axis and \(y\) is the position along the vertical axis.

In the context of sequences:
  • The x-axis typically represents the independent variable, \(n\), often indicating time or a step number.
  • The y-axis represents the dependent variable, \(a_n\), which is the sequence value corresponding to each \(n\).
  • For the sequence \(a_{n} = \frac{3n^{4} + n - 1}{5n^{4} + 2n^{2} + 1}\), points are plotted as \((n, a_{n})\).
This system provides a clear visual backdrop for analyzing how sequence terms change relative to \(n\). The coordinates simplify tracking specific terms, especially when determining trends or behaviors like limits or oscillations.
Sequence Behavior
Sequence behavior refers to how the terms of a sequence change as the index \(n\) increases. By graphing a sequence, one can observe these changes more clearly. There are several types of behaviors that sequences typically exhibit:
  • Increasing: The values of the sequence grow larger with each successive term.
  • Decreasing: The values get smaller as \(n\) increases.
  • Oscillating: Values fluctuate between a set of numbers, forming a pattern.
  • Converging: The sequence approaches a particular value, known as its limit.
By plotting the sequence \(a_{n} = \frac{3n^{4} + n - 1}{5n^{4} + 2n^{2} + 1}\), one can better determine which behavior the sequence follows. Observations from such graphs are crucial for understanding mathematical theories and predictions about sequences.
Limits of Sequences
A limit of a sequence refers to the value that the terms of a sequence approach as the index \(n\) approaches infinity. Not all sequences have limits, but identifying them is key in calculus and higher mathematics.

For the given sequence \(a_{n} = \frac{3n^{4} + n - 1}{5n^{4} + 2n^{2} + 1}\), as \(n\) grows larger, you may notice that the values of \(a_{n}\) approach a particular number. This number is its limit.

Consider these steps to find the limit:
  • Simplify the highest power of \(n\) in the numerator and denominator, in this case, both have \(n^4\).
  • Factor out \(n^4\) to observe the ratio of leading coefficients, \( \frac{3}{5} \).
  • As \(n\) tends towards infinity, terms with lower powers become negligible, emphasizing the importance of this ratio.
Thus, the sequence converges to the limit \( \frac{3}{5} \). Finding limits helps comprehend how sequences behave in the long-term and are pivotal in understanding their applications.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Beginning at 6: 45 A.M., a bus stops on my block every 23 minutes, so I used the formula for the \(n\) th term of an arithmetic sequence to describe the stopping time for the \(n\) th bus of the day.

55\. The probability that South Florida will be hit by a major hurricane (category 4 or 5) in any single year is \(\frac{1}{16}\). (Source: National Hurricane Center) a. What is the probability that South Florida will be hit by a major hurricane two years in a row? b. What is the probability that South Florida will be hit by a major hurricane in three consecutive years? c. What is the probability that South Florida will not be hit by a major hurricane in the next ten years? d. What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years?

What are mutually exclusive events? Give an example of two events that are mutually exclusive.

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins the jackpot by matching all five numbers drawn from white balls ( 1 through 56 ) and matching the number on the gold Mega Ball \(^{\oplus}\) ( 1 through 46 ). What is the probability of winning the jackpot?

a. If two people are selected at random, the probability that they do not have the same birthday (day and month) is \(\frac{365}{365} \cdot \frac{364}{365}\). Explain why this is so. (Ignore leap years and assume 365 days in a year.) b. If three people are selected at random, find the probability that they all have different birthdays. c. If three people are selected at random, find the probability that at least two of them have the same birthday. d. If 20 people are selected at random, find the probability that at least 2 of them have the same birthday. e. How large a group is needed to give a 0.5 chance of at least two people having the same birthday?
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