Understanding the Sequence 6, 12, 24, 48: Finding the Next Term

Introduction

Understanding patterns and sequences is a fundamental concept in mathematics, and it plays a significant role in various disciplines, including computer science, physics, and even in practical problem-solving situations. In this article, we will explore the pattern in the number sequence 6, 12, 24, 48, and determine the next term in the sequence.

This article will provide a comprehensive overview of the sequence, how to recognize the pattern, and how to find the next term using different approaches. We will also include practical examples and explanations to ensure clarity and understanding.

Detecting the Pattern: Doubling Each Step

The given sequence is 6, 12, 24, 48. Upon closer inspection, we can observe a simple and consistent pattern: each term is double the previous one. This type of progression is known as a Geometric Progression (GP) with a common ratio of 2.

What is a Geometric Progression?

A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the given sequence, the first term (a 6) and the common ratio (r 2).

The formula for the (n^{th}) term of a GP is given by:

(a_n a cdot r^{(n-1)})

Using this formula, we can derive any term in the sequence. For instance, to find the fourth term:

(a_4 6 cdot 2^{(4-1)} 6 cdot 2^3 6 cdot 8 48)

Verifying the Pattern: Doubling Each Step

Let's verify the pattern by doubling each term in the sequence to confirm the next number:

1st term: 6 2nd term: 6 * 2 12 3rd term: 12 * 2 24 4th term: 24 * 2 48 5th term: 48 * 2 96

This confirms that the next term in the sequence is 96.

Analyzing the Sequence Using Division Method

Another way to analyze the sequence is by using the division method. Dividing each term by the previous term gives a consistent ratio, indicating that the sequence is indeed following a doubling pattern.

Divide Each Successive Term

Let's divide each successive term to identify the common ratio:

24 / 6 4 24 / 12 2 48 / 12 4 96 / 48 2

Here, we observe that the ratio alternates between 2 and 4. However, when considering the doubling pattern, the consistent ratio is 2, confirming that the sequence follows a doubling progression.

Conclusion

In conclusion, the next term in the sequence 6, 12, 24, 48 is 96, as the pattern is clear: each term is double the previous term. Understanding such patterns is not only important for mathematical problem-solving but also for developing logical thinking skills.