Problem 94 - Prealgebra

Question

Find the next number in each of the geometric sequences below. \(1, \frac{1}{4}, \frac{1}{16}, \ldots\)

Step-by-Step Solution

Answer

The next number in the sequence is \( \frac{1}{64} \).

Step 1: Identify the Common Ratio

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio. To find the common ratio, divide the second term by the first term.So, our first term is 1, and our second term is \( \frac{1}{4} \).Common ratio, \( r = \frac{1/4}{1} = \frac{1}{4} \).

Step 2: Verify the Common Ratio

Verify that the common ratio is the same between the subsequent terms. Divide the third term by the second term: \[ \frac{1}{16} \div \frac{1}{4} = \frac{1}{16} \times \frac{4}{1} = \frac{1}{4} \].The common ratio is consistent.

Step 3: Calculate the Next Term

Using the common ratio, calculate the next term in the sequence. Multiply the last given term by the common ratio:Last given term = \( \frac{1}{16} \)Next term = \( \frac{1}{16} \times \frac{1}{4} = \frac{1}{64} \).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio plays a crucial role. It is the constant factor between consecutive terms. To determine the common ratio, we divide any term by the preceding term. For example, in the geometric sequence given, the common ratio can be found by dividing the second term by the first term, which results in \( \frac{1}{4} \).
This consistency makes the sequence predictable. Understanding the common ratio allows you to project future terms with accuracy.

• The common ratio is constant throughout the sequence.
• Dividing any term by its preceding term will give you the common ratio.
• A reliable tool for finding subsequent terms in the sequence.
• Essential for identifying whether a sequence is geometric or not.

Term Multiplication in Geometric Sequences

Once the common ratio is identified, term multiplication becomes a straightforward process. This involves multiplying a given term by the common ratio to find the next term.
For example, if we know that the common ratio is \( \frac{1}{4} \), multiplying any term in the sequence by \( \frac{1}{4} \) will yield the next term. This multiplication construction is key to extending geometric sequences further.

• Multiplying the last term by the common ratio extends the sequence.
• This method depends heavily on a consistent common ratio.
• Term multiplication is simple and eliminates complexities in sequences.
• Foundation for predicting terms far along the sequence without iteration.

Introduction to Mathematical Sequences

Mathematical sequences are ordered sets of numbers defined by specific rules. A geometric sequence is one type where each number, or term, is the result of multiplying the previous term by a fixed, non-zero number known as the common ratio.
This regularity sets geometric sequences apart from other sequences and enables the prediction of subsequent terms quickly. Understanding the nature of sequences gives insight into mathematical consistency and patterns.

• Sequences can be arithmetic, geometric, or neither.
• Patterns and predictability make sequences a fundamental mathematical concept.
• Recognizing the type of sequence helps in correctly applying rules like term multiplication.
• An essential framework for structuring numbers logically.