Find the wrong number in each of the following number series.
5 6 15 41 105 230 446.

Options

A. 105
B. 6
C. 230
D. 15

Answer

To find the wrong number in the series 5, 6, 15, 41, 105, 230, 446, we can analyze the pattern of the series step by step.

### Step-by-Step Solution:

1. **Identify the Series**: The given series is:
\[ 5, 6, 15, 41, 105, 230, 446 \]

2. **Calculate Differences**: We will find the differences between consecutive terms:
- \(6 - 5 = 1\)
- \(15 - 6 = 9\)
- \(41 - 15 = 26\)
- \(105 - 41 = 64\)
- \(230 - 105 = 125\)
- \(446 - 230 = 216\)

So, the differences are:
\[ 1, 9, 26, 64, 125, 216 \]

3. **Identify Patterns in Differences**: Now, we will look for a pattern in the differences:
- \(1 = 1^3\)
- \(9 = 2^3 + 1\)
- \(26 = 3^3 - 1\)
- \(64 = 4^3\)
- \(125 = 5^3\)
- \(216 = 6^3\)

We can see that the differences are related to cubes of integers, but with some variations.

4. **Check for Consistency**: The expected pattern seems to be:
- \(1^3\)
- \(2^3 + 1\)
- \(3^3 - 1\)
- \(4^3\)
- \(5^3\)
- \(6^3\)

However, we notice that the second term (15) does not fit the expected pattern.

5. **Propose a Correction**: If we replace 15 with a number that fits the pattern, we can check:
- If we replace 15 with 14, we can check:
- \(6 + 8 = 14\) (where \(8 = 2^3\))
- \(14 + 27 = 41\) (where \(27 = 3^3\))

This maintains the pattern of cubes.

6. **Conclusion**: The incorrect number in the series is 15. If we replace it with 14, the series follows the expected pattern.

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