Introduction

The series 2, 4, 6, 5, 7, 9, 8 has intrigued many as it seems to follow a pattern that is not immediately obvious. This article will delve into the pattern and help you understand how to determine the next number in the series.

Understanding the Pattern

The series starts with an ascending pattern of adding 2 to the previous number (2 to 4, 4 to 6, and so on). However, the pattern temporarily shifts to a descending sequence for a moment before returning to the ascending pattern.

Mathematical Representation

Let's break down the series and identify the structure:

2, 4, 6: Here, each number is 2 more than the previous number. 6, 5, 7: The sequence shifts to a descending order of 2 for a moment, then back to an ascending order. 7, 9, 8: After the temporary dip, the series resumes the pattern of adding 2.

Given this, the next number in the sequence would be:

8 2 10 10 2 12 12 - 1 11 (to follow the same descending pattern) 11 2 13 13 2 15 15 2 17 17 2 19 19 2 21

Thus, the next number is 21.

Verification Using Mathematical Formula

To further verify, we can use the arithmetic progression formula to predict the next number:

Given the last number in the series is 18, the common difference (d) is 2.

The formula for the next term (an) in an arithmetic sequence is:

an a1 (n-1)d

Here:

a1 (first term) 2 d (common difference) 2 n (term position) 8 (since 18 is the 8th term)

Let's find the 9th term (a9):

a9 2 (9-1) * 2 2 8 * 2 2 16 18 2 20

This confirms our earlier pattern analysis and the next number in the series is 20.

Additional Insights

If the series were to continue:

20 2 22 22 - 1 21 21 2 23 23 2 25 25 - 1 24

Based on the alternating pattern of ascending and descending by 1 every two terms, the next number after 21 would be 22.

Conclusion

The next number in the series 2, 4, 6, 5, 7, 9, 8 is 20, as it follows the established pattern of alternating between ascending and descending by 1.