What Will Be the Missing Number in the Sequence 15, 45, and 135?

Whenever a number sequence is presented, particularly in the realm of mathematics, it often ignites a spark of curiosity and challenges logical thinking. Among such puzzles, the sequence 15, 45, and 135 is a classic example that many people might attempt to solve. In this article, we will explore the logic behind this series and attempt to determine the next number in the sequence. We will delve into the characteristics of both arithmetic and geometric progressions, and see which one fits our sequence best.

Understanding the Sequence: An Analysis of 15, 45, and 135

Let's first break down the sequence: 15, 45, and 135. A quick glance at these numbers may lead some to believe the next number should be 90, based on a simple multiplication approach. Indeed, many people have proposed that 15×6 90, and reasoning that 15×3 45 and 15×9 135, this seems plausible. However, is 90 the right answer? Let's investigate further.

Arithmetic Progression: A Step-by-Step Method

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the preceding term. However, if we use this method with our numbers:

15 ? 45 45 ? 135

The differences between the terms do not remain consistent, indicating that our sequence is not an arithmetic progression.

Geometric Progression: The Secret to the Sequence

Of greater relevance to this sequence is the geometric progression (GP), where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the ratio. Let's examine how this fits our sequence.

15, 45, 135. Each term increases by multiplying the previous one by 3:

15 × 3 45 45 × 3 135

The pattern here is clear, and it suggests that we should multiply the last term, 135, by 3 to find the next term in the sequence.

135 × 3 405

Conclusion: Determining the Missing Number

Based on the geometric progression, the next number in the sequence 15, 45, and 135 would be 405, and not 90. As such, the complete sequence is: 15, 45, 135, 405.

While some might propose other methods, such as adding numbers or employing other arithmetic operations, the correct approach here is the geometric progression. It is both elegant in its simplicity and consistent in its application to the sequence.

Further Exploration: Other Mathematical Concepts

To deepen our understanding of mathematical sequences, let's briefly explore other concepts that might be related to number sequences:

Fibonacci Sequence: This is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. This sequence is unique and does not apply to our sequence of 15, 45, and 135. Lucas Sequence: Similar to the Fibonacci sequence, but begins with 2 and 1. For example, 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123. Again, this is not relevant to our sequence. Prime Number Sequence: A sequence of numbers where each number is a prime number, such as 2, 3, 5, 7, 11, 13. This sequence does not apply to 15, 45, and 135, as none of these numbers are prime.

While there are numerous number sequences to explore, the geometric progression in our sequence 15, 45, and 135 offers a clear and logical pattern that supports the answer of 405.

Final Thoughts: A Gateway to Mathematical Reasoning

The investigation into the sequence 15, 45, and 135 is a great way to exercise our mathematical reasoning and introduces us to the fascinating world of number sequences. By understanding the difference between arithmetic and geometric progressions, we can more accurately solve similar problems and enhance our logical thinking skills. Whether you are a student, a teacher, or anyone interested in mathematics, mastering these concepts can provide a strong foundation for more complex mathematical problems in the future.