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Chapter 9: Problem 45

Write an expression for the apparent \(n\) th term \(\left(a_{n}\right)\) of the sequence. (Assume that \(n\) begins with 1.) $$1,-1,1,-1,1, \ldots$$

Short Answer

Expert verified
The expression for the apparent nth term \(a_n\) of the sequence is \(-1^{n+1}\).

Step by step solution

01

Defining the Sequence

We can start by defining the sequence as \(a_1 = 1\), \(a_2 = -1\), \(a_3 = 1\), \(a_4 = -1\), and so on. In each case, the term alternates between -1 and 1.
02

Identify the Pattern

Looking at the sequence, we can see that the pattern of the terms is alternating between 1 and -1. This is a common property of sequences erected by the power of -1.
03

Formulate using powers of -1

Using the power of -1, we can represent the nth term of the sequence, \(a_n\), as \(-1^{n+1}\). When n is odd, \(n+1\) gives an even number, so \(-1^{n+1}\) will be 1. When n is even, \(n+1\) gives an odd number, so \(-1^{n+1}\) will be -1. This matches the existing sequence.

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Key Concepts

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

Alternating Sequences
To understand alternating sequences, picture a stoplight switching between red and green. In mathematics, an alternating sequence flips between two different values with each term. The sequence provided, \(1, -1, 1, -1, 1, \ldots\), is a basic example of an alternating sequence. Each term is the opposite of the term before it.

When dealing with alternating sequences, we often utilize terms that involve the powers of -1 to generate the alternation, since \( (-1)^n \) flips signs based on whether \( n \) is even or odd. Alternating sequences like these abound in different areas of mathematics and physics, where they model phenomena that switch between two states.
Powers of -1
Diving deeper into the powers of -1, it's fundamental to note that this concept helps express alternating sequences succinctly. Remember that any even power of -1 equals 1 (\( (-1)^2 = 1 \) or \( (-1)^4 = 1 \) ), and any odd power equals -1 (\( (-1)^1 = -1 \) or \( (-1)^3 = -1 \) ).

Thus, by adjusting the exponent, we produce the alternation in our sequence. If the nth term is represented as \( (-1)^{n+1} \) as in the textbook example, the addition of 1 ensures that the first term starts as positive. The power of -1 is a powerful tool in sequences, as it can capture the pattern of alternation with a simple, elegant expression.
Arithmetic Sequences
On the other hand, arithmetic sequences are quite different from the alternating sequence you've seen. An arithmetic sequence progresses by a constant difference, also known as the common difference. For example, in the sequence \(2, 4, 6, 8, \ldots\), the common difference is 2. Each term is created by adding 2 to the previous term.

An arithmetic sequence is generally denoted as \( a_n = a_1 + (n - 1)d \) where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. Unlike the alternating sequence, which is defined by the sign-changing behavior of terms, arithmetic sequences show a consistent growth or decrease throughout the terms.

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Most popular questions from this chapter

Solve for \(n\) $$_{n} P_{6}=12 \cdot_{n-1} P_{5}$$

American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered \(1-36,\) of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on three consecutive spins.

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes: "success" or "failure." The probability of a success on each trial is \(p,\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment.You toss a fair coin seven times. To find the probability of obtaining four heads, evaluate the term $$_{7} C_{4}\left(\frac{1}{2}\right)^{4}\left(\frac{1}{2}\right)^{3}$$ in the expansion of \(\left(\frac{1}{2}+\frac{1}{2}\right)^{7}\).

Prove the identity. \(_{n} P_{n-1}=_{n} P_{n}\)

Evaluate \(_{n} C_{r}\) using a graphing utility. \(_{50} C_{6}\)
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