Identifying the Last Number in the Series: 3 4 6 10 18 34 66 130

The series 3 4 6 10 18 34 66 130 has garnered attention for its unique pattern. Let's explore the logic behind the numbers and how to determine the next term in the sequence.

Pattern Recognition

The key to understanding this series lies in the differences between consecutive terms. By examining these differences, we can uncover the underlying mathematical pattern.

Let's calculate and observe the differences:

Differences Analysis

Step-by-Step Calculation

Starting with the series:

3 4 6 10 18 34 66 130

The differences between consecutive terms are:

4 - 3 1 6 - 4 2 10 - 6 4 18 - 10 8 34 - 18 16 66 - 34 32 130 - 66 64

We notice that these differences follow a pattern: 1, 2, 4, 8, 16, 32, 64. These are the powers of 2: 20, 21, 22, 23, 24, 25, 26.

Continuing the Pattern

Following this pattern, the next term in the differences sequence would be:

27 128

To find the next number in the series, we add this difference to the last term:

130 128 258

Thus, the next number in the series is 258.

Alternative Approaches

Maths-x suggests a different approach by observing that each term is obtained by doubling the previous term's difference:

3 to 4: 1

4 to 6: 2

6 to 10: 4

10 to 18: 8

18 to 34: 16

34 to 66: 32

66 to 130: 64

Following this pattern, the next term is:

130 to 258: 128

Additional Series for Reference

Let's also examine a related series: 6 15 35 77 143 _. By comparing the terms, we can identify the pattern:

15 6 x 23

35 15 x 25

77 35 x 27

143 77 x 29

Following this sequence, the next term would be:

143 x 211 357

357 x 213 727

Conclusion

The series 3 4 6 10 18 34 66 130 follows a specific pattern of doubling the difference between consecutive terms. Additionally, the pattern can be identified by the differences being powers of 2. By applying this understanding, we can easily determine the next number in the series, which is 258.

Understanding such patterns not only enhances our problem-solving skills but also broadens our approach to more complex mathematical concepts.