The next number in the series 2, 5, 11, 23 is 47.
Understanding Thabit Numbers
The given sequence, 2, 5, 11, 23, is a part of what's known as Thabit numbers (also sometimes called "Thabit's numbers"). These numbers follow a specific pattern, and recognizing this pattern allows us to predict the next number in the series.
Thabit Number Pattern
- Thabit numbers are generated by the formula: 3 2n - 1, where n* is a non-negative integer (0, 1, 2, 3...).
- The sequence begins when n = 1.
How the Sequence is Formed
Here's how the provided sequence is generated using the Thabit number formula:
| n | Formula (3 * 2n - 1) | Result |
|---|---|---|
| 1 | 3 21 - 1 = 3 2 - 1 | 5 |
| 2 | 3 22 - 1 = 3 4 - 1 | 11 |
| 3 | 3 23 - 1 = 3 8 - 1 | 23 |
| 4 | 3 24 - 1 = 3 16 - 1 | 47 |
As we can see, starting with n=1, these calculations correspond to the sequence 5, 11, 23... To get to the first number of the series i.e. 2, n must be 0. Therefore, the correct formula should be:
3 * 2n - 1 where n=0,1,2,3... This gives:
| n | Formula (3 * 2n - 1) | Result |
|---|---|---|
| 0 | 3 20 - 1 = 3 1 - 1 | 2 |
| 1 | 3 21 - 1 = 3 2 - 1 | 5 |
| 2 | 3 22 - 1 = 3 4 - 1 | 11 |
| 3 | 3 23 - 1 = 3 8 - 1 | 23 |
| 4 | 3 24 - 1 = 3 16 - 1 | 47 |
The Next Number
Using the same pattern, for n=4 we find that:
- 3 24 - 1 = 3 16 - 1 = 48 - 1 = 47
Therefore, the next number in the sequence is 47. The full list of Thabit numbers (Sequence A055010 in the OEIS) is: 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ...
