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Chapter 1: Problem 2

Find a formula for the \(n\)th term of the sequence. a. \(\\{3,3,3,3,3, \ldots\\}\) b. \(\\{1,4,16,64,256, \ldots\\}\) c. \(\left\\{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32} \ldots\right\\}\) d. \(\\{1,3,7,15,31, \ldots\\}\)

Short Answer

Expert verified
a. \( a_n = 3 \); b. \( a_n = 4^{n-1} \); c. \( a_n = \left( \frac{1}{2} \right)^n \); d. \( a_n = 2^n - 1 \).

Step by step solution

01

Analyze the Pattern of Sequence a

The sequence \( \{3, 3, 3, 3, 3, \ldots\} \) consists of the same number repeated. This suggests that the nth term of the sequence is constant, regardless of \( n \).
02

Formulate the nth Term for Sequence a

Since the pattern does not change, the formula for the nth term is simply \( a_n = 3 \) for all \( n \).
03

Identify the Pattern of Sequence b

Look at the sequence \( \{1, 4, 16, 64, 256, \ldots\} \). Notice that each term is a power of 4: \( 1 = 4^0, 4 = 4^1, 16=4^2, 64=4^3\), and so on.
04

Formulate the nth Term for Sequence b

The pattern suggests each term is \( 4^{n-1} \). Therefore, the nth term formula for this sequence is \( a_n = 4^{n-1} \).
05

Recognize the Pattern of Sequence c

This sequence \( \left\{ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \ldots \right\} \) shows that each term is a power of \( \frac{1}{2} \): \( \frac{1}{2} = \left( \frac{1}{2} \right)^1, \frac{1}{4} = \left( \frac{1}{2} \right)^2, \ldots \).
06

Derive the nth Term for Sequence c

In Sequence c, the pattern suggests the nth term is \( \left( \frac{1}{2} \right)^n \). Thus, the formula for the nth term is \( a_n = \left( \frac{1}{2} \right)^n \).
07

Examine the Pattern of Sequence d

For the sequence \( \{1, 3, 7, 15, 31, \ldots\} \), notice that each term \( a_n \) seems to be one less than a power of 2: \( 1 = 2^1 - 1, 3 = 2^2 - 1, 7 = 2^3 - 1, \ldots\).
08

Formulate the nth Term for Sequence d

Each term is described by the formula \( a_n = 2^n - 1 \). Therefore, the nth term for this sequence is \( a_n = 2^n - 1 \).

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Key Concepts

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

constant sequence
A constant sequence is a very straightforward type of sequence where every term in the sequence is the same. In mathematical terms, this means that for any position in the sequence, the value of the term does not change. This kind of sequence can be described by the general formula:
  • \( a_n = c \), where \( c \) is the constant value of every term in the sequence.
In our exercise, the sequence \( \{3, 3, 3, 3, 3, \ldots\} \) is a constant sequence because every term is 3. Hence, the formula for the nth term is simply \( a_n = 3 \). Constant sequences are easy to recognize and work with because of their simplicity; they play a fundamental role in understanding more complex sequences.
power sequence
A power sequence is characterized by each term being a power of a specific base number throughout the sequence. This type of sequence can always be expressed with a consistent base raised to a power. The formula typically looks like this:
  • \( a_n = b^{f(n)} \), where \( b \) is the base, and \( f(n) \) is some function of \( n \) that tells us the exponent.
For example, consider the sequence: \( \{1, 4, 16, 64, 256, \ldots\} \). Here, each term is a power of 4, as \( 4^0 = 1 \), \( 4^1 = 4 \), \( 4^2 = 16 \), etc. Hence, the nth term formula is \( a_n = 4^{n-1} \). In power sequences, understanding the base is crucial as it dictates the growth pattern of the sequence.
geometric sequence
A geometric sequence is a sequence in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula for the nth term of a geometric sequence typically includes the initial term and the common ratio:
  • \( a_n = a_1 \, r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
Taking the sequence \( \left\{ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \ldots \right\} \), each term is multiplied by the common ratio \( \frac{1}{2} \) to get to the next term. The nth term formula for this is \( a_n = \left( \frac{1}{2} \right)^n \). Recognizing this pattern helps in predicting future terms easily.
nth term derivation
Deriving the nth term formula for a sequence involves identifying a pattern or rule to describe the position of any term using a function of \( n \). This derivation is key to understanding and working with sequences, as it encapsulates the entire sequence in a single expression. To derive a formula:
  • Identify if the sequence is arithmetic, geometric, or another type.
  • Observe the relationship between consecutive terms.
  • Express the pattern you notice in terms of \( n \).
For example, consider the sequence \( \{1, 3, 7, 15, 31, \ldots\} \). Each term is one less than a power of 2, leading to the formula \( a_n = 2^n - 1 \). This allows any term in the sequence to be calculated without generating all prior terms. Understanding nth term derivation opens a world of possibilities in sequence prediction and analysis.

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Most popular questions from this chapter

In 1868 , the accidental introduction into the United States of the cottony- cushion insect (Icerya purchasi) from Australia threatened to destroy the American citrus industry. To counteract this situation, a natural Australian predator, a ladybird beetle (Novius cardinalis), was imported. The beetles kept the insects to a relatively low level. When DDT (an insecticide) was discovered to kill scale insects, farmers applied it in the hope of reducing the scale insect population even further. However, DDT turned out to be fatal to the beetle as well, and the overall effect of using the insecticide was to increase the numbers of the scale insect. Let \(C_{n}\) and \(B_{n}\) represent the cottony-cushion insect and ladybird beetle population levels, respectively, after \(n\) days. Generalizing the model in Problem 4, we have $$ \begin{aligned} &C_{n+1}=C_{n}+k_{1} C_{n}-k_{2} B_{n} C_{n} \\ &B_{n+1}=B_{n}-k_{3} B_{n}+k_{4} B_{n} C_{n} \end{aligned} $$ where the \(k_{i}\) are positive constants. a. Discuss the meaning of each \(k_{i}\) in the predator-prey model. b. What assumptions are implicitly being made about the growth of each species in the absence of the other species? c. Pick values for your coefficients and try several starting values. What is the longterm behavior predicted by your model? Vary the coefficients. Do your experimental results indicate that the model is sensitive to the coefficients? To the starting values? d. Modify the predator-prey model to reflect a predator-prey system in which farmers apply (on a regular basis) an insecticide that destroys both the insect predator and the insect prey at a rate proportional to the numbers present.

Build a numerical solution for the following initial value problems. Plot your data to observe patterns in the solution. Is there an equilibrium solution? Is it stable or unstable? a. \(a_{n+1}=-1.2 a_{n}+50, \quad a_{0}=1000\) b. \(a_{n+1}=0.8 a_{n}-100, \quad a_{0}=500\) c. \(a_{n+1}=0.8 a_{n}-100, \quad a_{0}=-500\) d. \(a_{n+1}=-0.8 a_{n}+100, \quad a_{0}=1000\) e. \(a_{n+1}=a_{n}-100, \quad a_{0}=1000\)

The data in the accompanying table show the speed \(n\) (in increments of \(5 \mathrm{mph}\) ) of an automobile and the associated distance \(a_{n}\) in feet required to stop it once the brakes are applied. For instance, \(n=6\) (representing \(6 \times 5=30 \mathrm{mph}\) ) requires a stopping distance of \(a_{6}=47 \mathrm{ft}\). a. Calculate and plot the change \(\Delta a_{n}\) versus \(n\). Does the graph reasonably approximate a linear relationship? b. Based on your conclusions in part (a), find a difference equation model for the stopping distance data. Test your model by plotting the errors in the predicted values against \(n .\) Discuss the appropriateness of the model. $$ \begin{array}{l|llllllllllllllll} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline a_{n} & 3 & 6 & 11 & 21 & 32 & 47 & 65 & 87 & 112 & 140 & 171 & 204 & 241 & 282 & 325 & 376 \\ \hline \end{array} $$

You owe $$\$ 500$$ on a credit card that charges \(1.5 \%\) interest each month. You can pay $$\$ 50$$ each month with no new charges. What is the equilibrium value? What does the equilibrium value mean in terms of the credit card? Build a numerical solution. When will the account be paid off? How much is the last payment?

Sociologists recognize a phenomenon called social diffusion, which is the spreading of a piece of information, a technological innovation, or a cultural fad among a population. The members of the population can be divided into two classes: those who have the information and those who do not. In a fixed population whose size is known, it is reasonable to assume that the rate of diffusion is proportional to the number who have the information times the number yet to receive it. If \(a_{n}\) denotes the number of people who have the information in a population of \(N\) people after \(n\) days, formulate a dynamical system to approximate the change in the number of people in the population who have the information.
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