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Chapter 1: Problem 9

9\. $$1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \dots$$

Short Answer

Expert verified
The general term of the sequence is \(a_n = \frac{1}{2n-1}\).

Step by step solution

01

Identify the Pattern

The series provided is a sequence of fractions: \(1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \dots\). To identify the pattern, observe the denominators: \(1, 3, 5, 7, 9\). This is an arithmetic sequence of odd numbers starting from 1. Each subsequent term increases by 2.
02

General Term Expression

From the pattern of the denominators, we observe they are of the form \(2n - 1\). So, each term in the sequence can be expressed using the formula \(\frac{1}{2n-1}\), where \(n = 1, 2, 3,\) and so on.
03

Determine the General Term

To find any term in the sequence, substitute \(n\) into the expression from Step 2. Thus, the general term for the sequence is \(a_n = \frac{1}{2n-1}\).

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Key Concepts

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

Pattern Recognition
The exercise revolves around understanding and identifying patterns in sequences, which is essential for solving many mathematical problems. In this particular sequence, we are given a set of fractions starting from 1: \(1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}\). Observing these numbers closely, you can see that the numerators are consistently 1, while the denominators are in a separate sequence by themselves. Here's what to do:
  • Look at the denominators: 1, 3, 5, 7, 9, which forms an arithmetic sequence.
  • Notice that each term differs by 2 from the previous one, starting from 1.
  • This sequence is a well-known pattern 鈥� the odd numbers.
Once you identify this sequence as an arithmetic one with a common difference of 2, it becomes easier to predict future terms. Recognizing patterns like this one helps simplify mathematical processes and solve problems quickly.
General Term Formula
Once the pattern is recognized, the next step is to find a formula that represents this sequence. A general term formula allows you to calculate any term in the sequence without listing all the previous ones. For the current sequence, our focus is on the denominators which are characterized by the expression \(2n - 1\), where \(n\) starts at 1 and increases by 1 for each subsequent term:
  • For \(n = 1\), the denominator is \(2 \times 1 - 1 = 1\).
  • For \(n = 2\), the denominator is \(2 \times 2 - 1 = 3\).
  • Continuing this pattern confirms the denominators are indeed \(1, 3, 5, 7, 9\).
Thus, each fraction in the sequence is expressed as \(\frac{1}{2n-1}\). This formula compacts the infinite process of predicting sequence terms into a simple calculation. It is both efficient and powerful, saving you time and effort.
Sequence Analysis
Sequence analysis involves understanding the structure and characteristics of a sequence by applying the general term formula you have derived. For our exercise, the analysis means using \(a_n = \frac{1}{2n-1}\) to determine any term's value.
  • To find the 5th term, substitute \(n = 5\) in the formula, producing \(\frac{1}{2 \times 5 - 1} = \frac{1}{9}\).
  • For the 10th term, you'd calculate \(\frac{1}{2 \times 10 - 1} = \frac{1}{19}\).
This kind of analysis facilitates a deeper understanding of a sequence's behavior and evolution over time. Instead of calculating from scratch or verifying individual terms, a sequence analysis with the general term provides an immediate and reliable conclusion, catering to efficient problem-solving and analysis.

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