Understanding the Sequence: 3, 7, 15, 31, and 63

Patterns in number sequences can be fascinating and are often the subject of mathematical exploration. Let’s dive into the series 3, 7, 15, 31, and 63 and discover the underlying pattern that generates each subsequent number in the series.

Doubling and Adding One

The given series 3, 7, 15, 31, and 63 follows a pattern where each term is generated by doubling the previous term and then adding one. This pattern can be observed as follows:

Starting with 1: 1 x 2 2, and 2 1 3 3 x 2 6, and 6 1 7 7 x 2 14, and 14 1 15 15 x 2 30, and 30 1 31 31 x 2 62, and 62 1 63

So, following this pattern, the next number in the series is 63.

Generating the First 10 Terms with J Programming Language

In the J programming language, generating the first 10 terms in the series can be done using the command /32^2 to 10. The output is as follows:

3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047

As you can see, the next term after 31 is indeed 63.

Manually Calculating the Differences

The differences between consecutive terms in the sequence can also reveal the pattern:

Difference between 7 and 3: 7 – 3 4 Difference between 15 and 7: 15 – 7 8 Difference between 31 and 15: 31 – 15 16

These differences are powers of 2, specifically 2^2, 2^3, and 2^4. The next difference in the sequence would be 2^5 32. Adding 32 to the last number in the sequence, 31, gives us 63:

31 32 63

Another Method to Generate the Sequence

Another method to generate the sequence is by starting from the first term:

First term: 3 Second term: 7, which is 3 times 2 plus 1 (3 x 2 1 7) Third term: 15, which is 7 times 2 plus 1 (7 x 2 1 15) Fourth term: 31, which is 15 times 2 plus 1 (15 x 2 1 31) Following this pattern: the next term would be 31 times 2 plus 1 (31 x 2 1 63)

Known Sequence and Its Reference

This sequence is known as the sequence of numbers that are one less than a power of 2. Specifically, each term in the sequence is of the form 2^n - 1. The sequence can be further explored by noting that the differences between consecutive terms are always powers of 2, as observed earlier. Therefore, the next number in the sequence is 63.

The pattern can be summarized as 3, 7, 15, 31, 63, with each term being one less than a power of 2. Understanding these patterns is valuable for both mathematical exploration and algorithm design.