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Chapter 10: Problem 14

Find a formula for the \(n\) th term of the sequence. The sequence \(-1,1,-1,1,-1, \ldots\)

Short Answer

Expert verified
The nth term formula is \(a_n = (-1)^n\).

Step by step solution

01

Identify the Pattern

Observe the given sequence: \(-1, 1, -1, 1, -1, \ldots\). Notice that the terms alternate between \(-1\) and \(1\). This indicates that the sequence alternates sign as it progresses.
02

Determine the General Formula

The terms alternate between sign, which can be represented using a power of \(-1\). For odd \(n\): \((-1)^{n} = -1\).For even \(n\): \((-1)^{n} = 1\).Therefore, a suitable formula for the nth term \(a_n\) is \(a_n = (-1)^n\).

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Key Concepts

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

Alternating Sequence
An alternating sequence is a list of numbers where each term changes sign. This means it switches between positive and negative values.
In our example with the sequence \(-1, 1, -1, 1, -1, \ldots\), it's clear that the sequence switches from \(-1\) to \(1\) and back again.
  • The terms of this type of sequence flip signs in a consistent manner, creating a recognizable pattern.
  • This pattern is repeated throughout the sequence, helping us predict future terms.
Recognizing this type of pattern is essential for understanding how the sequence behaves and is key to identifying the nth term formula.
nth Term Formula
The nth term formula is a mathematical expression that allows us to calculate any term in a sequence without listing all previous terms.
For alternating sequences, like our example, the formula can be represented as \((-1)^n\).
  • When \(n\) is odd, \((-1)^n\) results in \(-1\), giving us the negative term of the sequence.
  • When \(n\) is even, \((-1)^n\) equals \(1\), producing the positive term.
This formula simplifies the process of finding terms in a sequence and highlights the orderly nature of mathematical patterns within sequences.
Mathematical Pattern Recognition
Mathematical pattern recognition involves identifying regularities or trends in numbers. When looking at sequences, recognizing these patterns enables us to find general formulas.
In our sequence \(-1, 1, -1, 1, -1, \ldots\), examining how the signs of terms alternate is key.
  • By spotting that the sign flips with every term, we recognize the sequence follows a recurring pattern.
  • This allows for the deduction of an overall rule or formula that describes the sequence, like \(a_n = (-1)^n\).
Understanding these patterns not only aids in deriving formulas but also in predicting future terms or solving similar problems with other sequences.

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Most popular questions from this chapter

Linearizations at inflection points Show that if the graph of a twice- differentiable function \(f(x)\) has an inflection point at \(x=a,\) then the linearization of \(f\) at \(x=a\) is also the quadratic approximation of \(f\) at \(x=a\). This explains why tangent lines fit so well at inflection points.

The series $$\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\frac{x^{9}}{9 !}-\frac{x^{11}}{11 !}+\cdots$$ converges to \(\sin x\) for all \(x\) a. Find the first six terms of a series for \(\cos x\). For what values of \(x\) should the series converge? b. By replacing \(x\) by \(2 x\) in the series for \(\sin x,\) find a series that converges to \(\sin 2 x\) for all \(x\) c. Using the result in part (a) and series multiplication, calculate the first six terms of a series for \(2 \sin x \cos x .\) Compare your answer with the answer in part (b).

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{n}=n \sin \frac{1}{n}\)

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{n}=\frac{8^{n}}{n !}\)

A cubic approximation Use Taylor's formula with \(a=0\) and \(n=3\) to find the standard cubic approximation of \(f(x)=\) \(1 /(1-x)\) at \(x=0 .\) Give an upper bound for the magnitude of the error in the approximation when \(|x| \leq 0.1\)
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