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Chapter 14: Problem 9

Consider the sequence \(0.3,0.33\) \(0.333, \dots\) What is the seventh term of this sequence?

Short Answer

Expert verified
The seventh term is 0.3333333.

Step by step solution

01

Identify the Pattern

Observe that each term in the sequence adds an extra '3' after the decimal point. This is an increasing sequence of decimal representations of fractions with a continuously repeating digit '3' after the decimal point.
02

Determine the General Term Formula

The n-th term of the sequence is constructed by having n '3's after the decimal point. The nth term can be expressed as \(0.\underbrace{333\cdots3}_{n \text{ times}}\).
03

Calculate the Seventh Term

Following the pattern from the earlier steps, the seventh term is the number 0. with seven '3's after the decimal point: \(0.3333333\).

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Key Concepts

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

Pattern Recognition in Decimal Sequences
Recognizing patterns in a sequence is like piecing together a puzzle. For the sequence given: \(0.3, 0.33, 0.333, \ldots\), the pattern revolves around the number of \('3's\) after the decimal point. Every term you encounter adds one more \('3'\), making this sequence straightforward and predictable.
Pattern recognition helps us identify the consistent rule applied to generate the sequence. In this scenario, each step in the sequence modifies the prior term by appending an additional \('3'\) to its end.
  • The first term is \(0.3\), which is essentially \(0.\) followed by one \('3'\).
  • The second term, \(0.33\), has two \('3's\).
  • This pattern simply continues, where each subsequent term increases the number of \('3's\) by one.
Understanding this simple structure allows us to predict future elements of the sequence efficiently, without having to count manually each time.
Defining the Sequence Formula
Once a pattern is recognized, the next step involves defining a formula to easily find any term in the sequence. For our specific sequence, this is quite straightforward. The sequence formula hinges on the count of \('3's\) after the decimal point.
For the n-th term in this sequence, you'll write the number \('3'\), repeat it \(n\) times, and place them all after the zero and decimal point.
To put it mathematically:
  • The n-th term = \(0.\underbrace{333\cdots3}_{n \text{ times}}\), where \(n\) is the position of the term in the sequence.
This method allows us to express any term of the sequence compactly and makes it easier to compute higher terms without overly complex calculations.
Calculating Specific Terms in the Sequence
Armed with our sequence formula, calculating any particular term becomes a straightforward task. Suppose you're tasked with finding the seventh term of the sequence.
The instruction from our formula tells us to place seven \('3's\) after the decimal point. Thus, the seventh term is written as:
  • Seventh term = \(0.3333333\).
Each term highlights the power of using a pattern and formula: it turns what could be a tedious process into a quick, systematic approach. This not only enhances efficiency but also boosts our confidence in tackling larger sequences.

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