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Chapter 9: Problem 12

Write the first four terms of the sequence \(\left\\{a_{n}\right\\}_{n=1}^{\infty}.\) $$a_{n}=2+(-1)^{n}$$

Short Answer

Expert verified
$$ Answer: The first four terms of the sequence are 1, 3, 1, and 3.

Step by step solution

01

Calculate the 1st term#a_{1}

To find the first term, plug in n=1 into the given formula: $$a_{1}=2+(-1)^{1}$$ $$a_{1}=2-1$$ $$a_{1}=1$$
02

Calculate the 2nd term#a_{2}

To find the second term, plug in n=2 into the given formula: $$a_{2}=2+(-1)^{2}$$ $$a_{2}=2+1$$ $$a_{2}=3$$
03

Calculate the 3rd term#a_{3}

To find the third term, plug in n=3 into the given formula: $$a_{3}=2+(-1)^{3}$$ $$a_{3}=2-1$$ $$a_{3}=1$$
04

Calculate the 4th term#a_{4}

To find the fourth term, plug in n=4 into the given formula: $$a_{4}=2+(-1)^{4}$$ $$a_{4}=2+1$$ $$a_{4}=3$$ The first four terms of the sequence \(\left\\{a_{n}\right\\}_{n=1}^{\infty}\) are 1, 3, 1, and 3.

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Key Concepts

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

Arithmetic
Arithmetic sequences are one of the fundamental types of sequences in mathematics. In an arithmetic sequence, the difference between consecutive terms is constant, known as the common difference. However, the sequence given in the exercise\( \{a_{n}\}_{n=1}^{\infty} \) is **not** an arithmetic sequence. This is because the difference between consecutive terms alternates rather than remains constant.

In arithmetic sequences, you would typically find each term by adding the common difference to the previous term. However, it's essential to recognize when a sequence does not follow this pattern, as it indicates that other factors, such as an alternating pattern, may be in play.
Alternating Patterns
Alternating patterns in sequences can create a valuable, yet sometimes confusing, variation that differs from linear or arithmetic trends. In the provided problem, the formula \( a_{n} = 2 + (-1)^{n} \) creates a sequence where terms switch back and forth due to the \((-1)^{n}\) component.

  • When \( n \) is odd, \((-1)^{n}\) resolves to \(-1\), resulting in terms of 1.
  • When \( n \) is even, \((-1)^{n}\) equals \(+1\), resulting in terms of 3.

This alternating change causes the sequence to oscillate between two values, which is neither increasing nor decreasing consistently. Understanding this helps in predicting future terms once you identify the pattern. Recognizing the pattern type guides students in setting expectations for what each term will be, based on its position in the sequence.
Formula Evaluation
Evaluating a sequence formula involves substituting values into an expression to find specific terms. For the sequence \( \{a_{n}\}_{n=1}^{\infty} \), you apply the given formula \( a_{n} = 2 + (-1)^{n} \) to compute each term by replacing \( n \) with the desired term number.

Here's how to evaluate this specific formula:
  • **Calculate Term 1**: Set \( n = 1 \). The equation becomes \( a_{1} = 2 + (-1)^{1} = 1 \).
  • **Calculate Term 2**: Set \( n = 2 \). The equation becomes \( a_{2} = 2 + (-1)^{2} = 3 \).
  • **Calculate Term 3**: Set \( n = 3 \). The equation becomes \( a_{3} = 2 + (-1)^{3} = 1 \).
  • **Calculate Term 4**: Set \( n = 4 \). The equation becomes \( a_{4} = 2 + (-1)^{4} = 3 \).

This method shows how substituting numerical values into an algebraic formula yields specific terms in the sequence, demonstrating the pattern and aiding in the understanding of its behavior.

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Most popular questions from this chapter

a. Consider the number 0.555555...., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 \ldots\) b. Consider the number \(0.54545454 \ldots,\) which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots \ldots, n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form of the number. d. Try the method of part (c) on the number \(0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 3^{-k}$$

Showing that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \operatorname{In} 1734,\) Leonhard Euler informally proved that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} .\) An elegant proof is outlined here that uses the inequality $$ \cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x\left(\text { provided that } 0

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine a plausible value for the limit of \(\left\\{S_{n}\right\\}.\) $$h_{0}=20, r=0.75$$

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\) b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\) c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\)
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