52, 54, 60, 84, 132, 212, 332

Options

A. 52
B. 60
C. 84
D. 54

Explanation

To determine the odd one out in the series: 52, 54, 60, 84, 132, 212, 332, we will analyze the differences between the consecutive numbers and identify any patterns.

### Step-by-Step Solution:

1. **Identify the Series**:
The given series is:
\[ 52, 54, 60, 84, 132, 212, 332 \]

2. **Calculate the Differences**:
We will calculate the differences between consecutive terms:
- \(54 - 52 = 2\)
- \(60 - 54 = 6\)
- \(84 - 60 = 24\)
- \(132 - 84 = 48\)
- \(212 - 132 = 80\)
- \(332 - 212 = 120\)

So, the differences are:
\[ 2, 6, 24, 48, 80, 120 \]

3. **Analyze the Differences**:
Next, we will look for a pattern in the differences:
- The differences are:
\[ 2, 6, 24, 48, 80, 120 \]
We can see that these numbers can be related to squares of odd numbers:
- \(2 = 3^2 - 7\)
- \(6 = 3^2 - 3\)
- \(24 = 5^2 - 1\)
- \(48 = 7^2 - 1\)
- \(80 = 9^2 - 1\)
- \(120 = 11^2 - 1\)

4. **Identify the Pattern**:
The pattern can be observed as:
- The differences can be expressed as \(n^2 - 1\) where \(n\) is an odd number.
- The expected differences should follow the pattern of squares of odd numbers decreasing by 1.

5. **Check the Expected Differences**:
- The expected difference after \(52\) should be \(3^2 - 1 = 8\).
- If we add \(8\) to \(52\), we get \(60\), which is correct.
- The next difference should be \(5^2 - 1 = 24\) (which is correct).
- The next should be \(7^2 - 1 = 48\) (which is correct).
- The next should be \(9^2 - 1 = 80\) (which is correct).
- The last should be \(11^2 - 1 = 120\) (which is correct).

6. **Identify the Odd One Out**:
Since the first number \(52\) is correct, but \(54\) does not fit the pattern as it should be \(52 + 8 = 60\), we can conclude that:
- The odd one out in the series is \(54\).

### Conclusion:
The odd one out in the series is **54**.