Next number of the sequence 2 9 28 65 126 is

Options

A. 195
B. 199
C. 208
D. 217

Answer

To find the next number in the sequence 2, 9, 28, 65, 126, we will analyze the differences between consecutive terms and identify a pattern.

**Step 1: List the given sequence.**
The sequence is: 2, 9, 28, 65, 126.

**Step 2: Calculate the differences between consecutive terms.**
- Difference between 9 and 2: 9 - 2 = 7
- Difference between 28 and 9: 28 - 9 = 19
- Difference between 65 and 28: 65 - 28 = 37
- Difference between 126 and 65: 126 - 65 = 61

So, the first differences are: 7, 19, 37, 61.

**Step 3: Calculate the differences of the first differences (second differences).**
- Difference between 19 and 7: 19 - 7 = 12
- Difference between 37 and 19: 37 - 19 = 18
- Difference between 61 and 37: 61 - 37 = 24

So, the second differences are: 12, 18, 24.

**Step 4: Calculate the differences of the second differences (third differences).**
- Difference between 18 and 12: 18 - 12 = 6
- Difference between 24 and 18: 24 - 18 = 6

So, the third differences are: 6, 6.

**Step 5: Identify the pattern in the third differences.**
The third differences are constant (6). This suggests that the next second difference will also increase by 6.

**Step 6: Calculate the next second difference.**
The next second difference after 24 will be: 24 + 6 = 30.

**Step 7: Calculate the next first difference.**
Add the next second difference to the last first difference:
Next first difference = 61 + 30 = 91.

**Step 8: Calculate the next term in the sequence.**
Add the next first difference to the last term in the sequence:
Next term = 126 + 91 = 217.

Thus, the next number in the sequence is **217**.

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