in the following number series only one is wrong . Find out the wrong number .
`32 " "34 " "37" "46 " "62 " "87" " 123`

Options

A. 34
B. 37
C. 62
D. 87

Answer

To solve the problem of identifying the wrong number in the series `32, 34, 37, 46, 62, 87, 123`, we will follow these steps:

### Step 1: Identify the Differences
First, we will calculate the differences between consecutive numbers in the series.

- Difference between 34 and 32: \( 34 - 32 = 2 \)
- Difference between 37 and 34: \( 37 - 34 = 3 \)
- Difference between 46 and 37: \( 46 - 37 = 9 \)
- Difference between 62 and 46: \( 62 - 46 = 16 \)
- Difference between 87 and 62: \( 87 - 62 = 25 \)
- Difference between 123 and 87: \( 123 - 87 = 36 \)

So, we have the differences: \( 2, 3, 9, 16, 25, 36 \).

### Step 2: Analyze the Differences
Next, we will analyze the differences we calculated:

- The differences are: \( 2, 3, 9, 16, 25, 36 \).
- Notice that these differences can be related to perfect squares:
- \( 2 \) can be seen as \( 1^2 + 1 \)
- \( 3 \) can be seen as \( 2^2 - 1 \)
- \( 9 \) is \( 3^2 \)
- \( 16 \) is \( 4^2 \)
- \( 25 \) is \( 5^2 \)
- \( 36 \) is \( 6^2 \)

### Step 3: Identify the Pattern
The differences appear to follow a pattern of squares:
- The differences should be \( 1^2, 2^2, 3^2, 4^2, 5^2, 6^2 \).

### Step 4: Find the Missing Number
Following the pattern, the differences should be:
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)

The sequence of differences should be \( 1, 4, 9, 16, 25, 36 \).

If we replace the second number (34) with 33:
- The new differences would be:
- \( 33 - 32 = 1 \)
- \( 37 - 33 = 4 \)
- \( 46 - 37 = 9 \)
- \( 62 - 46 = 16 \)
- \( 87 - 62 = 25 \)
- \( 123 - 87 = 36 \)

This confirms that the correct sequence of differences follows the perfect squares.

### Conclusion
Thus, the wrong number in the series is **34**, and it should be replaced with **33**.

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