Find the wrong number in the following number series:
750 535 411 348 322 314 315

Options

A. 315
B. 750
C. 411
D. 314

Answer

To find the wrong number in the series: 750, 535, 411, 348, 322, 314, 315, we will analyze the differences between consecutive numbers and look for a pattern.

### Step-by-Step Solution:

1. **Identify the Series**: The series given is:
\[ 750, 535, 411, 348, 322, 314, 315 \]

2. **Calculate the Differences**:
- From 750 to 535:
\[ 750 - 535 = 215 \]
- From 535 to 411:
\[ 535 - 411 = 124 \]
- From 411 to 348:
\[ 411 - 348 = 63 \]
- From 348 to 322:
\[ 348 - 322 = 26 \]
- From 322 to 314:
\[ 322 - 314 = 8 \]
- From 314 to 315:
\[ 314 - 315 = -1 \]

3. **List the Differences**:
The differences are:
\[ 215, 124, 63, 26, 8, -1 \]

4. **Identify a Pattern**:
The differences appear to be decreasing in a specific pattern. Let's analyze:
- \(215\) is close to \(216\) (which is \(6^3\))
- \(124\) is close to \(125\) (which is \(5^3\))
- \(63\) is close to \(64\) (which is \(4^3\))
- \(26\) is close to \(27\) (which is \(3^3\))
- \(8\) is close to \(9\) (which is \(2^3\))
- \(-1\) is close to \(0\) (which is \(1^3\))

5. **Establish the Pattern**:
It seems that we are subtracting \(n^3 - 1\) from the previous number:
- For \(n=6\): \(216 - 1 = 215\)
- For \(n=5\): \(125 - 1 = 124\)
- For \(n=4\): \(64 - 1 = 63\)
- For \(n=3\): \(27 - 1 = 26\)
- For \(n=2\): \(8 - 1 = 8\)
- For \(n=1\): \(1 - 1 = 0\)

6. **Find the Missing Number**:
Continuing the pattern, we should have:
- From \(322\), we should subtract \(2^3 - 1 = 7\):
\[ 322 - 7 = 315 \]
This means that instead of \(314\), we should have \(315\) in the series.

7. **Conclusion**:
The wrong number in the series is:
\[ \text{Wrong number: } 314 \]