1, 1, 3, 9, 6, 36, 10, 100, …, 225

Options

A. 15
B. 16
C. 20
D. 25

Explanation

To solve the series: 1, 1, 3, 9, 6, 36, 10, 100, …, 225, we can break it down into two separate sequences.

### Step-by-Step Solution:

1. **Identify the Pattern**:
We can separate the numbers into two sequences:
- First sequence (above): 1, 3, 6, 10, ?
- Second sequence (below): 1, 9, 36, 100, 225

2. **Analyze the First Sequence**:
- The first sequence is: 1, 3, 6, 10, ?
- Let's find the differences between consecutive terms:
- 3 - 1 = 2
- 6 - 3 = 3
- 10 - 6 = 4
- The differences are increasing by 1 each time (2, 3, 4).
- Following this pattern, the next difference should be 5.
- Therefore, the next term will be:
- 10 + 5 = 15.

3. **Analyze the Second Sequence**:
- The second sequence is: 1, 9, 36, 100, 225
- Let's see if there's a pattern:
- 1 = 1^2
- 9 = 3^2
- 36 = 6^2
- 100 = 10^2
- 225 = 15^2
- The numbers being squared are: 1, 3, 6, 10, 15.
- These numbers are the triangular numbers, which correspond to the first sequence.

4. **Conclusion**:
- The missing number in the first sequence is 15.
- Therefore, the complete series is: 1, 1, 3, 9, 6, 36, 10, 100, 15, 225.

### Final Answer:
The missing number in the series is **15**.