Problem 51 - Algebra 2

Question

a. Show that the infinite geometric series \(0.142857+\) \(0.000000142857+\ldots\) has a sum of \(\frac{1}{7}\) b. Find the fraction form of the repeating decimal 0.428571428571\(\ldots\)

Step-by-Step Solution

Answer

The sum of the given infinite geometric series is \(\frac{1}{7}\) and the fraction form of the repeating decimal 0.428571428571 is \(\frac{3}{7}\).

Step 1: Identify the common ratio for the geometric series

Using the equation \(a + ar + ar^2 + \ldots\), identify the base 'a' and common ratio 'r'. Here, \(a=0.142857\) and \(r=0.000001\).

Step 2: Calculate the sum using infinite geometric series formula

The formula for the sum of an infinite geometric series is \(S = \frac{a}{1-r}\). Substitute values \(a=0.142857\) and \(r=0.000001\) to get the sum.

Step 3: Evaluate and verify the result

On substituting \(a=0.142857\) and \(r=0.000001\) in \(S = \frac{a}{1-r}\) we get \(S = \frac{1}{7}\). So the infinite geometric series sums up to \(\frac{1}{7}\).

Step 4: Identify the repeating pattern in the given decimal

For the second part, the repeating pattern in the decimal 0.428571428571 is '428571'. Let's denote the decimal fraction as 'x'. So, \(x=0.428571428571\).

Step 5: Convert the repeating decimal into a fraction

To convert the repeating decimal to a fraction, the formula to use is \(x = \frac{a}{999999}\), where 'a' is the repeated six-digit pattern '428571'. So, \(x = \frac{428571}{999999}\).

Step 6: Simplify the fraction

On simplifying the fraction, we get \(x = \frac{3}{7}\). So, fraction form of the repeating decimal 0.428571428571 is \(\frac{3}{7}\).

Key Concepts

Understanding the Common Ratio

In an infinite geometric series, the common ratio, denoted by \( r \), is a crucial element. It tells us how each term is related to the previous one. The series follows the pattern \( a + ar + ar^2 + \, \ldots \), where \( a \) is the first term and \( r \) is the constant that each term is multiplied by to get the next term.

• For the series \( 0.142857 + 0.000000142857 + \ldots \), the first term \( a \) is \( 0.142857 \).
• The common ratio \( r \) is \( 0.000000142857 / 0.142857 \), which computes to \( 0.000001 \).

Even though \( r \) seems tiny in this case, it plays a big role in determining the sum of the series. A common ratio between 0 and 1 ensures that the sum of the infinite series converges, meaning it approaches a finite number. This is why we can calculate the sum of such infinite series using the formula \( S = \frac{a}{1-r} \), where \( S \) is the sum of the series.

Deciphering Repeating Decimals

Repeating decimals are decimals that have one or more repeating digits or sequences of digits after the decimal point. This periodic nature can also be denoted using a line (for example, \( 0.428571\overline{428571} \)) or with ellipses (\( 0.428571428571\ldots \)).

• To work with repeating decimals, identify the repeating block. For example, in \( 0.428571428571 \ldots \), the repeating block is '428571'.
• This is essential for converting repeating decimals into fractions, as the repeating digit pattern informs the calculation process.

Understanding the repeating pattern is key to deciphering and calculating these types of numbers. This allows us to transition from a seemingly complex number into a simpler fractional form.

Converting a Repeating Decimal to a Fraction

Transforming a repeating decimal into a fraction can seem challenging, but there's a structured process to simplify it. Let's take the decimal \( 0.428571428571\ldots \) as an example.

• Identify the repeating sequence: It is '428571', comprising six digits.
• To convert this, set up the equation \( x = 0.428571428571... \).
• The next step is to express this as \( x = \frac{a}{999999} \), where \( a \) represents the digits '428571'.
• In this format, \( x = \frac{428571}{999999} \).
• Finally, simplify the fraction by dividing both numerator and denominator by their greatest common divisor to reach the simplest form, which is \( \frac{3}{7} \).

This conversion method uses the repetitiveness and structure of decimals to derive a neat and manageable fraction. Such techniques help simplify long decimals and comprehend their equivalent fractional values.