Problem 56 - College Algebra with Modeling and Visualization

Question

Write each national number in the form of an infinite geometric series. $$\frac{23}{99}$$

Step-by-Step Solution

Answer

The number \(\frac{23}{99}\) is an infinite geometric series: \(0.23 + 0.23 \times 0.01 + 0.23 \times 0.01^2 + ... \).

Step 1: Recognize the Decimal Representation

The fraction \( \frac{23}{99} \) is a repeating decimal. To find the decimal form, divide 23 by 99, which gives 0.232323... with "23" repeating indefinitely.

Step 2: Express the Repeating Decimal

The repeating decimal 0.232323... can be expressed as 0.23 + 0.0023 + 0.000023 + ... Each term in this sequence is obtained by multiplying the previous term by 0.01.

Step 3: Identify the Common Ratio

Observe that each subsequent term in the series is 0.01 times the previous term. Thus, the common ratio \( r \) of this geometric series is 0.01.

Step 4: Write the Infinite Geometric Series

The decimal 0.232323... can be represented as an infinite geometric series \( 0.23 + 0.23 imes 0.01 + 0.23 imes 0.01^2 + ... \). This showcases the series where each consecutive term is 0.23 multiplied by increasing powers of 0.01.

Key Concepts

Repeating Decimal

A repeating decimal is a decimal number that continues forever while repeating a pattern of digits. In the case of the fraction \( \frac{23}{99} \), when we divide 23 by 99, the result is a decimal where the digits "23" repeat indefinitely.

• The process of converting a fraction to its decimal form involves division, which in this case results in 0.232323... .
• This repeating sequence of '23' highlights that after a certain point, the decimal pattern begins again, infinitely.

With repeating decimals, it's important to not only recognize the pattern but also understand how these can be translated into other mathematical forms, such as a geometric series.

Common Ratio

The common ratio is a key component of a geometric series, representing how each term in the series relates to the one before it. When dealing with repeating decimals like 0.232323..., the decimal can be expressed in terms of a geometric series.

• In the series for 0.232323..., each subsequent term is formed by multiplying the number before it by 0.01.
• For instance, after starting with 0.23, multiplying by 0.01 gives us the next term: 0.0023.

This repetitive multiplication by 0.01 designates 0.01 as the common ratio \( r \). Understanding the common ratio is crucial as it helps us to foresee how each term will evolve, growing smaller at a predictable rate as the sequence progresses.
This consistent rate of change is what allows a repeating decimal to transform into a geometric series conceptually.

Geometric Series Representation

A geometric series is a sequence where each term after the first is found by multiplying the previous term by a constant, known as the common ratio. When representing a repeating decimal like 0.232323... as a geometric series, we want to capture the structure of its repetition.

• The initial term of our series is 0.23.
• Subsequent terms are obtained by multiplying this first term by the common ratio, 0.01.
• This creates an infinite series: \( 0.23 + 0.23 \times 0.01 + 0.23 \times 0.01^2 + \ldots \).

Each term grows successively smaller because we are continuously multiplying by the common ratio, further showing how the decimal repeats. Writing 0.232323... this way gives a deeper understanding of its structure, transforming a straightforward repeating decimal into an informative mathematical series that can be analyzed and summed.