Unraveling the Pattern: What Comes Next in the Sequence 12.5, 25, 37.5, 50?

When dealing with numerical sequences, spotting the underlying pattern is key to predicting the next term. This article delves into a particular sequence and explains how to identify and extend it.

Understanding the Sequence

The given sequence is: 12.5 25 37.5 50

Each term follows a simple rule: each subsequent term is obtained by adding 12.5 to the previous term. Let's break this down:

Pattern Analysis

25 - 12.5 12.5 37.5 - 25 12.5 50 - 37.5 12.5

This consistent difference of 12.5 confirms the arithmetic progression. An arithmetic sequence is defined as a series of numbers in which the difference between consecutive terms is constant.

Finding the Next Term

Continuing the Pattern

To find the next term in the sequence, we simply follow the rule of adding 12.5 to the last given term:

50 12.5 62.5

Here is the sequence with the next term added:

12.5 25 37.5 50 62.5

Generalizing the Rule

The rule for this arithmetic sequence can be expressed as:

an a1 (n - 1)d

Where: an the nth term a1 the first term d the common difference (12.5 in this case) n the term number

Example Calculation

For the 5th term (n 5): a5 12.5 (5 - 1) * 12.5 12.5 4 * 12.5 12.5 50 62.5

Practical Applications

Mathematical Problem-Solving

Understanding and identifying patterns in sequences is crucial in mathematical problem-solving. It involves critical thinking, a solid grasp of arithmetic, and the ability to apply formulas to solve real-world problems.

Real-World Scenarios

In finance, understanding such patterns can help predict trends in stock prices or interest rates. In science, analyzing patterns in data can lead to groundbreaking discoveries. In everyday life, recognizing patterns helps in budgeting and financial planning.

Conclusion

Recognizing and extending arithmetic sequences is a valuable skill in various fields, from mathematics to finance and beyond. By understanding the rule governing the sequence 12.5, 25, 37.5, 50, we can confidently predict the next term as 62.5. This knowledge not only enhances problem-solving abilities but also provides insights into the patterns that underpin our world.

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