The missing numbers in the sequence are 3 and 99, forming a complete sequence of 3, 15, 35, 63, 99.
Understanding the Pattern
The provided reference states: "Hence, we get the final series as 3, 15, 35, 63, 99, 143". This indicates the full series. Let's analyze how this sequence is constructed.
The Series
The sequence is: 3, 15, 35, 63, 99, 143
Pattern Identification
- Differences: If we subtract each number from the next, we do not get a consistent difference. Instead, we find that the differences themselves have a pattern:
- 15 - 3 = 12
- 35 - 15 = 20
- 63 - 35 = 28
- 99 - 63 = 36
- 143 - 99 = 44
- Second Differences: The differences of differences are a constant 8:
- 20 - 12 = 8
- 28 - 20 = 8
- 36 - 28 = 8
- 44 - 36 = 8
This reveals that this is a quadratic sequence, which can also be expressed as the formula n n + (2 n) 2, where n represents the position of the number in the sequence starting with n=1. More concisely, this is (2n - 1) (2n + 1)
- n=1: (21 - 1) (21 + 1) = 1 3 = 3
- n=2: (22 - 1) (22 + 1) = 3 5 = 15
- n=3: (23 - 1) (23 + 1) = 5 7 = 35
- n=4: (24 - 1) (24 + 1) = 7 9 = 63
- n=5: (25 - 1) (25 + 1) = 9 11 = 99
- n=6: (26 - 1) (26 + 1) = 11 13 = 143
Completing the Sequence
Based on our analysis, the sequence 3, 15, 35, 63, 99 has 3 and 99 as the missing numbers when starting with the initial sequence of 15, 35, 63.
| Position (n) | Calculation | Result |
|---|---|---|
| 1 | (2 1 - 1) (2 * 1 + 1) | 3 |
| 2 | (2 2 - 1) (2 * 2 + 1) | 15 |
| 3 | (2 3 - 1) (2 * 3 + 1) | 35 |
| 4 | (2 4 - 1) (2 * 4 + 1) | 63 |
| 5 | (2 5 - 1) (2 * 5 + 1) | 99 |
Conclusion
The complete sequence is 3, 15, 35, 63, 99. Therefore the missing numbers in the sequence are 3 and 99.
