1,2,6, 15,31,56,91

Options

A. 31
B. 91
C. 56
D. 15

Answer

To find the odd number in the series 1, 2, 6, 15, 31, 56, 91, we will analyze the pattern of differences between the consecutive terms.

### Step-by-Step Solution:

1. **List the Series**:
The given series is:
\[ 1, 2, 6, 15, 31, 56, 91 \]

2. **Calculate the Differences**:
We will find the differences between consecutive terms:
- \(2 - 1 = 1\)
- \(6 - 2 = 4\)
- \(15 - 6 = 9\)
- \(31 - 15 = 16\)
- \(56 - 31 = 25\)
- \(91 - 56 = 35\)

So the differences are:
\[ 1, 4, 9, 16, 25, 35 \]

3. **Identify the Pattern in Differences**:
The differences \(1, 4, 9, 16, 25\) are perfect squares:
- \(1 = 1^2\)
- \(4 = 2^2\)
- \(9 = 3^2\)
- \(16 = 4^2\)
- \(25 = 5^2\)

The next perfect square would be \(6^2 = 36\).

4. **Check the Last Difference**:
The last difference we calculated was \(35\), but according to the pattern, it should be \(36\).

5. **Identify the Odd Number**:
Since the last term \(91\) was derived using \(35\) instead of \(36\), we need to find the correct term that should follow \(56\) if we were to add \(36\):
\[ 56 + 36 = 92 \]

Therefore, \(91\) is the odd number in the series as it does not fit the pattern.

### Conclusion:
The odd number in the series is **91**.