Introduction

Series and sequences are fundamental topics in mathematics, often challenging but intriguing for enthusiasts and students alike. This article dives into a specific example to illustrate how to determine the next number in a sequence, like 58, 56, 52, 46, 38. We will explore different methods and verify the next number in the series using both arithmetic and quadratic patterns.

Arithmetic Pattern: The Most Obvious Approach

Let's start by analyzing the given sequence: 58, 56, 52, 46, 38. To simplify, we observe the differences between consecutive terms:

Differences: 58 - 56 2 56 - 52 4 52 - 46 6 46 - 38 8

The differences form a sequence of even numbers: 2, 4, 6, 8, and so on. This pattern suggests that the next difference could be 10. Applying this reasoning, we can find the next number in the original series:

38 - 10 28

Thus, the next number is 28.

General Rules and Higher-Degree Patterns

While the arithmetic pattern provides a clear answer, sequences can follow more complex rules. One such rule involves a quadratic sequence, where the rate of change between differences is constant. Let's explore how the difference of differences is constant:

4 - 2 2 6 - 4 2 8 - 6 2

This constant second difference indicates a quadratic sequence. The general form for a quadratic sequence is given by:

[ t_n an^2 bn c ]

Using the first five terms, we can solve for coefficients (a), (b), and (c).

General Term Formula

Given the term (t_n), we can use the following steps:

Calculate the second differences, which are constant for a quadratic sequence. Here, the second differences are 2. Use the formula for the difference's difference: [ 2a 2 Rightarrow a 1 ] Using (a 1), we can find (b) by using the first term differences: [ 2(1)(1) b - 2 2 Rightarrow 2 b - 2 2 Rightarrow b 2 ] Finally, we can solve for (c) using the first term of the sequence: [ 2(1)^2 2(1) c 58 Rightarrow 2 2 c 58 Rightarrow c 54 ]

Thus, the formula for the sequence is:

[ t_n n^2 2n 54 ]

Substituting (n 6), we get:

[ t_6 6^2 2(6) 54 36 12 54 28 ]

This confirms that the next number in the sequence is indeed 28.

Conclusion

Both methods confirm that the next number in the sequence 58, 56, 52, 46, 38 is 28. Whether through an arithmetic pattern or a quadratic formula, this example demonstrates how understanding the underlying sequence can help predict the next term accurately.

Keywords: number sequence, quadratic sequence, arithmetic pattern

Note: The J Programming language code provided can be used to generate more terms of the series:

/58-2*i.10
58 56 52 46 38 28 16 2 _14 _32 _52

This confirms that the next term after 38 is 28.