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Chapter 2: Problem 43

Use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=\frac{1}{4} a_{n}+1\) and \(a_{0}=0\), find \(a_{14}\).

Short Answer

Expert verified
The value of \(a_{14}\) is approximately 1.333.

Step by step solution

01

Initialize Parameters

Start by understanding that the first term, \(a_0\), is given as 0. The recursive formula we need to use is \(a_{n+1} = \frac{1}{4}a_n + 1\). Our goal is to find the 15th term, \(a_{14}\).
02

Set Up Spreadsheet

In your spreadsheet, label the first column as 'n' indicating the term number, and the second column as 'a_n' indicating the value of the sequence at that term. In the first row, input 0 for 'n' and 0 for 'a_0'.
03

Apply Recursive Formula in Spreadsheet

In the second row of your spreadsheet, for the 'n' column enter 1, and for the 'a_n' column enter the formula \(= \text{(cell containing } a_0 \text{)}/4 + 1\). This will calculate the next term, \(a_1\).
04

Fill Down Formula

Drag the formula from Step 3 down the 'a_n' column until you reach \(n=14\). Each cell in this column should reference the cell above it, applying \(a_{n+1} = \frac{1}{4}a_n + 1\).
05

Read the Result

Once all computations are complete, the cell at \(n=14\) will contain the value of \(a_{14}\), which is the term you need to find.

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Key Concepts

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

Spreadsheet Calculations
To effectively calculate recursively defined sequences and manage data, spreadsheets offer a powerful tool. They allow you to automate repetitive tasks, such as calculating sequence terms, by setting up a systematic approach with formulas. A spreadsheet is particularly useful when working with sequences because it lets you easily apply a formula to a range of cells and update terms with minimal manual effort.
  • Start by opening a spreadsheet software like Microsoft Excel or Google Sheets.
  • Label columns clearly to track terms and their values.
  • Input initial values and set cells to reference one another, enabling forward computation via formulas.
  • Use drag-and-fill techniques to automate the application of recursive formulas down columns.
Overall, using a spreadsheet not only enhances accuracy but also simplifies the computation of complex sequences. This ensures that even if you're calculating up to the 14th term or beyond, the process remains clear and manageable.
Sequence Terms
In sequences, each term presents a unique position and value within a list. A term in a sequence is typically represented by a subscript, such as \(a_n\), where \(n\) specifies the term's position.
The position tells you where the term appears, and the value is what you calculate or assign based on the sequence's rule. For instance, in the context of recursive sequences, you're provided with an initial term, which serves as a starting point for calculating subsequent terms according to a specific formula.
  • Begin with the initial term, such as \(a_0\) which might be provided, here it's 0.
  • Follow the recursive relationship to find later terms, such as \(a_1, a_2, ..., a_{14}\).
  • The formula allows each term to be calculated based on the previous term.
Understanding the position and value of each sequence term is crucial because it enables you to identify patterns or predict future values using consistent rules.
Recursive Formula Application
Recursive formulas define each term in a sequence based on one or more preceding terms. This ties the sequence together, creating a chain of calculations that depend on prior terms for determining subsequent values. In our given exercise, the recursive rule is \(a_{n+1} = \frac{1}{4}a_n + 1\).
When applying this:
  • Start with known values: Here, \(a_0 = 0\).
  • The formula specifies how to determine each subsequent term \(a_{n+1}\) based on \(a_n\).
  • For instance, \(a_1 = \frac{1}{4}a_0 + 1\) results in \(a_1 = 1\) since \(\frac{1}{4} imes 0 + 1 = 1\).
  • Each application of the formula is contingent on correctly computing the prior term.
This recursive application necessitates careful attention to detail; an error in one term can cascade through subsequent calculations.

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Most popular questions from this chapter

Because of complex interactions with other drugs, some drugs have zeroth order elimination kinetics in some circumstances, and first order kinetics in other circumstances, depending on what other drugs are in the patient's system, as well as on age and preexisting medical conditions. Use the data on how concentration varies with time to determine whether the drug has zeroth or first order kinetics. Given the following sequence of measurements for drug concentration, determine whether the drug has zeroth or first order kinetics. $$ \begin{array}{lcccc} \hline \boldsymbol{t} \text { (Hours) } & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{c}_{t} \text { (mg/liter) } & 16 & 12 & 9 & 6.75 \\ \hline \end{array} $$

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{9}{a_{n}}\right), a_{0}=-1 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=2, x_{0}=0.2\)

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=4-2 a_{n}, a_{0}=\frac{4}{3} $$

Mountain Gorilla Conservation You are trying to build a mathematical model for the size of the population of mountain gorillas in a national park in Uganda. The data in this question are taken from Robbins et al. (2009). (a) You start by writing a word equation relating the population of gorillas \(t\) years after the study begins, \(N_{t}\), to the population \(N_{t+1}\) in the next year: \(N_{t+1}=N_{t}+\begin{array}{l}\text { number of gorillas } \\ \text { born in one year }\end{array}-\begin{array}{l}\text { number of gorillas } \\\ \text { that die in one yeai }\end{array}\) We will derive together formulas for the number of births and the number of deaths. (i) Around half of gorillas are female, \(75 \%\) of females are of reproductive age, and in a given year \(22 \%\) of the females of reproductive age will give birth. Explain why the number of births is equal to: \(\quad 0.5 \cdot 0.75 \cdot 0.22 \cdot N_{t}=0.0825 \cdot N_{t}\) (ii) In a given year \(4.5 \%\) of gorillas will die. Write down a formula for the number of deaths. (iii) Write down a recurrence equation for the number of gorillas in the national park. Assuming that there are 300 gorillas initially (that is \(N_{0}=300\) ), derive an explicit formula for the number of gorillas after \(t\) years. (iv) Calculate the population size after \(1,2,5\), and 10 years. (v) According to the model, how long will it take for population size to double to 600 gorillas? (b) In reality the population size is almost totally stagnant (i.e., \(N_{t}\) changes very little from year to year). Robbins et al. (2009) consider three different explanations for this effect: (i) Increased mortality: Gorillas are dying sooner than was thought. What percentage of gorillas would have to die each year for the population size to not change from year to year? (ii) Decreased female fecundity: Gorillas are having fewer offspring than was thought. Calculate the female birth rate (percentage of reproductive age females that give birth) that would lead to the population size not changing from year to year. Assume that all other values used in part (a) are correct. (iii) Emigration: Gorillas are leaving the national park. What number of gorillas would have to leave the national park each year for the population to not change from year to year?
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