What is the next term in the following sequence
2 3 11 38 102 ?

Options

A. 225
B. 227
C. 230
D. 235

Answer

To find the next term in the sequence 2, 3, 11, 38, 102, we will analyze the differences between consecutive terms and look for a pattern.

### Step-by-Step Solution:

1. **List the Sequence**:
The given sequence is:
\[ 2, 3, 11, 38, 102 \]

2. **Calculate the Differences**:
We will calculate the differences between consecutive terms:
- \(3 - 2 = 1\)
- \(11 - 3 = 8\)
- \(38 - 11 = 27\)
- \(102 - 38 = 64\)

So, the first differences are:
\[ 1, 8, 27, 64 \]

3. **Calculate the Differences of the Differences**:
Now, we will calculate the differences of the first differences:
- \(8 - 1 = 7\)
- \(27 - 8 = 19\)
- \(64 - 27 = 37\)

So, the second differences are:
\[ 7, 19, 37 \]

4. **Calculate the Differences of the Second Differences**:
Next, we calculate the differences of the second differences:
- \(19 - 7 = 12\)
- \(37 - 19 = 18\)

So, the third differences are:
\[ 12, 18 \]

5. **Calculate the Differences of the Third Differences**:
Finally, we calculate the differences of the third differences:
- \(18 - 12 = 6\)

So, the fourth difference is:
\[ 6 \]

6. **Identify the Pattern**:
The differences suggest a cubic pattern since the fourth differences are constant. The first differences \(1, 8, 27, 64\) correspond to \(1^3, 2^3, 3^3, 4^3\). The next term should be \(5^3 = 125\).

7. **Calculate the Next Term**:
To find the next term in the original sequence, we add the next difference (125) to the last term (102):
\[ 102 + 125 = 227 \]

Thus, the next term in the sequence is:
\[ \boxed{227} \]