Problem 55 - College Algebra with Modeling and Visualization

Question

Write each national number in the form of an infinite geometric series. $$ \frac{1}{7} $$

Step-by-Step Solution

Answer

\( \frac{1}{7} = 0.142857 + 0.000142857 + 0.000000142857 + \ldots \)

Step 1: Understand the Problem

We need to express \( \frac{1}{7} \) as an infinite geometric series, meaning we need a sequence whose sum equals \( \frac{1}{7} \), where each term in the series is derived from the previous one by multiplying it by a constant ratio.

Step 2: Identify the Decimal Representation

First, convert \( \frac{1}{7} \) into its decimal form using division. The decimal representation of \( \frac{1}{7} \) is 0.142857142857... which repeats every six digits.

Step 3: Recognize the Repeating Block

The sequence '142857' is a repeating block. This provides us with the terms of the geometric series necessary to express \( \frac{1}{7} \) as an infinite series. Each cycle of '142857' represents part of the sum.

Step 4: Express as a Series in Decimals

Using the repeating block '142857', we can express \( \frac{1}{7} \) as an infinite series: \(0.142857 + 0.000142857 + 0.000000142857 + \ldots\).

Step 5: Convert Decimals to Fractions

Recognize that \(0.142857 = \frac{142857}{10^6}\). Therefore, the series can be written as \( \frac{142857}{10^6} + \frac{142857}{10^{12}} + \frac{142857}{10^{18}} + \ldots \).

Step 6: Define the Geometric Series

The series identified is geometric, with the first term \( a = \frac{142857}{10^6} \) and the common ratio \( r = \frac{1}{10^6} \). Thus, the sum of the series is \( \frac{a}{1-r} \) which simplifies back to \( \frac{1}{7} \).

Key Concepts

Geometric Series

A geometric series is a sum of terms in which each term is a fixed multiple, known as the common ratio, of the previous term. This type of series can be finite or infinite. In mathematical terms, a geometric series is expressed as: \[ a + ar + ar^2 + ar^3 + ext{...} \] where \( a \) is the first term, and \( r \) is the common ratio. For it to be infinite, like in the case of \( \frac{1}{7} \), the series continues indefinitely. When \( |r| < 1 \), the sum of the infinite series is given by the formula: \[ \frac{a}{1-r} \] This means that if you can express a number as an infinite sum where each term is a multiple of the previous, you're dealing with a geometric series. In this exercise, the infinite geometric series lets us convert the fraction \( \frac{1}{7} \) into a repeating decimal through a sequence of numbers.

Repeating Decimals

Repeating decimals are decimals in which one or more digits repeat infinitely. These often arise when you convert fractions into decimals that cannot terminate. For instance, when we convert \( \frac{1}{7} \), we see the repeating sequence "142857". In such sequences:

• The digits that repeat are called the repeating block.
• The length of this block tells us how far apart the repeating parts occur.

Understanding repeating decimals is crucial in recognizing patterns that indicate they can be expressed as fractions. This concept also implies that every repeating decimal can be expressed by a geometric series, where each repeating cycle contributes incrementally to the overall sum.

Fraction to Decimal Conversion

Converting fractions to decimals is a fundamental skill in mathematics. This conversion helps understand the nature of the fraction and its decimal counterpart. For instance, to convert the fraction \( \frac{1}{7} \) into a decimal, you use long division. This can reveal whether the decimal representation terminates or repeats. For fractions like \( \frac{1}{7} \), the process shows a repeating decimal sequence in the result. The repeating pattern, "142857", helps identify:

• The position and length of the repeating block.
• The fractional representation of the repeating block in decimal form.

Knowing how to convert and understand the output, whether repeating or terminating, allows one to reverse the process, converting decimals back to fractions, similar to recognizing \( \frac{1}{7} \) as the infinite geometric series of its repeating block.