Problem 93 - Prealgebra

Question

Find the next number in each of the geometric sequences below. \(1, \frac{1}{3}, \frac{1}{9}, \ldots\)

Step-by-Step Solution

Answer

The next number in the sequence is \(\frac{1}{27}\).

Step 1: Identify the pattern of the sequence

To find the next number in the sequence, first identify the pattern. This is a geometric sequence, where each term is multiplied by the same number to get the next term.

Step 2: Determine the common ratio

Divide the second term by the first term to determine the common ratio, which is \(\frac{1}{3} \div 1 = \frac{1}{3}\). Check this by dividing the third term by the second term: \(\frac{1}{9} \div \frac{1}{3} = \frac{1}{3}\). This confirms the common ratio is \(\frac{1}{3}\).

Step 3: Calculate the next term

Multiply the last term provided by the common ratio to calculate the next term in the sequence: \(\frac{1}{9} \times \frac{1}{3} = \frac{1}{27}\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, each number after the first is found by multiplying the previous one by a fixed, non-zero number. This fixed number is called the common ratio. The common ratio can be discovered by dividing the second term by the first term of the sequence. For instance, if you have a sequence like \(1, \frac{1}{3}, \frac{1}{9}, \ldots\), you find the common ratio as follows:

• Divide \(\frac{1}{3}\) (the second term) by \(1\) (the first term) to get the common ratio, which is \(\frac{1}{3}\).
• Double-check this ratio by dividing \(\frac{1}{9}\) (the third term) by \(\frac{1}{3}\) (the second term). You'll also get \(\frac{1}{3}\), confirming it's consistent.

The common ratio tells us how rapidly the sequence grows or shrinks. In our example, since the ratio \(\frac{1}{3}\) is less than \(1\), the sequence is decreasing.

Following a Step by Step Solution for Geometric Sequences

Solving problems related to geometric sequences often requires a systematic approach. Here's a step by step solution:

• **Step 1:** Identify the type of sequence. Check if each term is being produced by multiplying the previous term by a consistent number. This verification will tell you it’s a geometric sequence.
• **Step 2:** Determine the common ratio by dividing consecutive terms. Only when all divisions yield the same result can you confirm the presence of a common ratio.
• **Step 3:** Calculate the next term by using the common ratio. Simply multiply the last known term by the common ratio to find the subsequent number.

This systematic approach ensures accuracy and simplifies the process of dealing with sequences. Such a methodical path reduces errors and increases understanding.

Importance of Pattern Recognition in Geometric Sequences

Recognizing patterns is fundamental when working with geometric sequences. Here’s why understanding patterns matters:

• Patterns help predict the subsequent terms of a sequence without recalculating from the start each time. Recognizing the pattern lets you apply the common ratio efficiently.
• By identifying patterns, students can grasp how sequences work in different contexts. This is crucial because it builds a strong mathematical intuition that aids in future, more complex topics.
• In the sequence \( 1, \frac{1}{3}, \frac{1}{9}, \ldots\), noticing the pattern allows us to foresee that each term is getting smaller by a factor of the common ratio, \(\frac{1}{3}\). Understanding this enables quick calculations of any next terms.

Pattern recognition goes beyond mathematical calculations; it strengthens cognitive abilities and problem-solving skills. The more you practice recognizing these patterns, the better you become at intuitively solving sequence-related problems.