To find the missing number in the sequence 2, 9, 20, 37, 64, 107, we can start by examining the differences between consecutive terms in the series.
Calculating the differences:
- 9 – 2 = 7
- 20 – 9 = 11
- 37 – 20 = 17
- 64 – 37 = 27
- 107 – 64 = 43
So, we have the differences: 7, 11, 17, 27, and 43. Let’s analyze the differences further:
Calculating the differences of these differences:
- 11 – 7 = 4
- 17 – 11 = 6
- 27 – 17 = 10
- 43 – 27 = 16
Now we have another series: 4, 6, 10, 16. Let’s find the differences once again:
- 6 – 4 = 2
- 10 – 6 = 4
- 16 – 10 = 6
Now we see a pattern in the second level differences: 2, 4, 6. These are increasing by 2. This suggests that the difference between the differences is following a quadratic sequence.
From the patterns and calculations, we can conclude that the sequence of the differences suggests that a number is missing between 20 and 37. If we find the next difference after 7, 11, we expect it to be around 17 which gives 20 + 17 = 37. But as we see there’s a jump, and we might find the middle number using 17 from the first series.
Let’s calculate the missing number:
To find the next number, we could say:
- 7 + 4 = 11 (next difference)
- 11 + 6 = 17
- 17 + 10 = 27
- 27 + 16 = 43
So, the missing number should be:
2, 9, 20, X, 37, 64, 107
Based on the pattern, we conclude that the missing number between 20 and 37 is 28.
Thus, the complete sequence is: 2, 9, 20, 28, 37, 64, 107.