What is the Next Number in this Sequence: 2, 5, 10, 17, 26, 37, ___?

The sequence 2, 5, 10, 17, 26, 37, ___ seems inconspicuous at first glance, but it is actually a carefully crafted mathematical puzzle. Let's explore the pattern and determine the next number.

The Sequence and Its Underlying Pattern

The sequence given is 2, 5, 10, 17, 26, 37. To find the next number, we must first identify the underlying pattern. One way to approach this is to look at the differences between consecutive terms:

5 - 2 3 10 - 5 5 17 - 10 7 26 - 17 9 37 - 26 11

We observe that the differences between consecutive terms are 3, 5, 7, 9, and 11, forming an arithmetic sequence. These differences are each 2 more than the previous one. If this pattern holds, the next difference would be 13.

Solving for the Next Term

Let's add the next difference to the last term in the sequence:

37 13 50

Thus, the next number in the sequence is 50. This approach provides a straightforward method to find the next term based on the observed differences.

Additional Insights: The Quadratic Sequence Form

A more mathematical approach involves recognizing the sequence as a quadratic sequence. The general form of a quadratic sequence is:

t_n an^2 bn c

To solve for the next term, we can use the differences to determine the coefficients:

The second difference in a quadratic sequence is constant. In this case, it is 2. The first difference gives us key information to solve for the linear coefficient. The initial term allows us to solve for the constant term.

By following these steps, we can verify that the next term indeed is 50. This method aligns with the previously mentioned approach of adding the next difference.

Connecting to Powers of 2: An Alternative Reality

Interestingly, if the sequence were based on the differences of increasing powers of 2 (1, 2, 4, 8, 16, etc.), the sequence would be entirely different. For instance, the sequence 33, 34, 35, 36, 37 would be expected. However, if the sequence were taken from the ideas in Douglas Adams' The Hitchhiker's Guide to the Galaxy, the next number might be 65 (2^6 64, and 65 is a play on the answer to life, the universe, and everything).

While the original sequence adheres to a consistent pattern, these alternatives add a layer of fun and creativity to the exploration of mathematical sequences.

Explore more on:

Quadratic Sequence Formula

By understanding these patterns and methods, you can decode intricate sequences and appreciate the beauty of mathematical puzzles.