1, 11, 38, 90, 175..

Options

A. 11
B. 78
C. 175
D. 301

Explanation

To find the next term in the series: 1, 11, 38, 90, 175, we can analyze the pattern in the differences between the terms.

### Step-by-Step Solution:

1. **Identify the terms in the series:**
- The series is: 1, 11, 38, 90, 175.

2. **Calculate the differences between consecutive terms:**
- 11 - 1 = 10
- 38 - 11 = 27
- 90 - 38 = 52
- 175 - 90 = 85

So, the first differences are: 10, 27, 52, 85.

3. **Calculate the differences of the first differences (second differences):**
- 27 - 10 = 17
- 52 - 27 = 25
- 85 - 52 = 33

So, the second differences are: 17, 25, 33.

4. **Calculate the differences of the second differences (third differences):**
- 25 - 17 = 8
- 33 - 25 = 8

The third differences are constant: 8.

5. **Predict the next second difference:**
- Since the third differences are constant, the next second difference will also be 33 + 8 = 41.

6. **Predict the next first difference:**
- The next first difference will be 85 + 41 = 126.

7. **Predict the next term in the series:**
- The next term will be 175 + 126 = 301.

Thus, the next term in the series is **301**.