5, 27, 61, 122, 213, 340, 509

Options

A. 27
B. 61
C. 122
D. 509

Explanation

To find the odd one out in the series: 5, 27, 61, 122, 213, 340, 509, we need to identify a pattern among these numbers.

### Step-by-Step Solution:

1. **Identify the Pattern**:
We will check if the numbers can be represented in terms of cubes of integers. The cubes of the first few integers are:
- \(1^3 = 1\)
- \(2^3 = 8\)
- \(3^3 = 27\)
- \(4^3 = 64\)
- \(5^3 = 125\)
- \(6^3 = 216\)
- \(7^3 = 343\)
- \(8^3 = 512\)

2. **Adjust the Cubes**:
Let's check if our series can be represented as cubes of integers minus 3:
- \(1^3 - 3 = 1 - 3 = -2\) (not in the series)
- \(2^3 - 3 = 8 - 3 = 5\) (matches the first number)
- \(3^3 - 3 = 27 - 3 = 24\) (not in the series)
- \(4^3 - 3 = 64 - 3 = 61\) (matches the third number)
- \(5^3 - 3 = 125 - 3 = 122\) (matches the fourth number)
- \(6^3 - 3 = 216 - 3 = 213\) (matches the fifth number)
- \(7^3 - 3 = 343 - 3 = 340\) (matches the sixth number)
- \(8^3 - 3 = 512 - 3 = 509\) (matches the seventh number)

3. **Identify the Odd One Out**:
From the above, we see that:
- The first number (5) corresponds to \(2^3 - 3\).
- The second number (27) does not fit the pattern as it should be 24.
- The third number (61) corresponds to \(4^3 - 3\).
- The fourth number (122) corresponds to \(5^3 - 3\).
- The fifth number (213) corresponds to \(6^3 - 3\).
- The sixth number (340) corresponds to \(7^3 - 3\).
- The seventh number (509) corresponds to \(8^3 - 3\).

Therefore, the number that does not fit the pattern is **27**.

### Conclusion:
The odd one out in the series is **27**.

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