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Chapter 11: Problem 21

Find the nth, or general, term. $$\frac{1}{x}, \frac{1}{x^{2}}, \frac{1}{x^{3}}, \ldots$$

Short Answer

Expert verified
\( \frac{1}{x^n} \)

Step by step solution

01

Identify the Pattern

Examine the sequence given: \( \frac{1}{x}, \frac{1}{x^2}, \frac{1}{x^3}, \ldots \). Each term in the sequence is in the form of a fraction where the numerator is always 1, and the denominator is a power of x. Notice that the power of x in the denominator increases by 1 with each term.
02

Express the Term Generically

To find the general term, express the nth term in terms of n. From the pattern observed, the first term has the denominator \(x^1\), the second term has \(x^2\), and the third term has \(x^3\). Thus, the nth term will have \(x^n\) in the denominator.
03

Write the General Term

Based on the observation, the general term (nth term) of the sequence can be expressed as \( \frac{1}{x^n} \).

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Key Concepts

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

Understanding Geometric Sequences
A geometric sequence is a type of sequence where each term is found by multiplying the previous term by a constant. This constant is called the common ratio. For instance, in the sequence given in the exercise: \( \frac{1}{x}, \frac{1}{x^2}, \frac{1}{x^3}, \ldots \), each term is obtained by multiplying the previous term by \( \frac{1}{x} \). The common ratio here is \( \frac{1}{x} \). Geometric sequences are highly regular and predictable, which makes them useful in various mathematical applications. Identifying the common ratio is key to understanding the pattern and behavior of the sequence.
Formulating the General Term
The general term (or nth term) of a sequence describes the position of any given term within the sequence based on its position number, n. In the exercise, we observed the pattern where the denominator of each term takes the form \( x^n \). The general term helps simplify the process of finding any term in the sequence without listing all preceding terms. To express the given sequence ( \( \frac{1}{x}, \frac{1}{x^2}, \frac{1}{x^3}, \ldots \) ) generically, we write the nth term as \( \frac{1}{x^n} \). This notation is valuable because it provides a direct formula for any term in the sequence when the term's position (n) is known.
Recognizing Mathematical Patterns
Mathematical patterns are regularly repeated arrangements that follow a specific rule or formula. Recognizing these patterns allows for the prediction of future terms and simplifies the problem-solving process. In the exercise above, the clear pattern is in how the power of the denominator increases linearly: 1, 2, 3, and so forth. By translating this pattern into a formula, \( \frac{1}{x^n} \), students can understand that each term's structure is consistent with the observed increase in the exponent. Identifying and understanding mathematical patterns enhances skills in differentiation and integration across various mathematical problems.

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