Find the wrong term in the following number series questions.
100,120,154,192,248,320,410

Options

A. 100
B. 248
C. 410
D. 154

Answer

To find the wrong term in the number series: 100, 120, 154, 192, 248, 320, 410, we will analyze the differences between consecutive terms and identify any inconsistencies.

### Step 1: Write down the series
The series given is:
- 100, 120, 154, 192, 248, 320, 410

### Step 2: Calculate the differences between consecutive terms
Let's find the differences between each pair of consecutive terms:
- 120 - 100 = 20
- 154 - 120 = 34
- 192 - 154 = 38
- 248 - 192 = 56
- 320 - 248 = 72
- 410 - 320 = 90

So, the differences are:
- 20, 34, 38, 56, 72, 90

### Step 3: Calculate the differences of the differences
Next, we will find the differences of these differences:
- 34 - 20 = 14
- 38 - 34 = 4
- 56 - 38 = 18
- 72 - 56 = 16
- 90 - 72 = 18

So, the second level of differences is:
- 14, 4, 18, 16, 18

### Step 4: Analyze the second differences
Now, we will analyze the second differences to see if there is a consistent pattern:
- The differences are inconsistent, indicating a potential issue with one of the original terms.

### Step 5: Identify the wrong term
Looking closely at the original differences, we notice that the jump from 120 to 154 is larger than expected. If we replace 154 with 150, we can check the new differences:
- 100, 120, 150, 192, 248, 320, 410

Calculating the new differences:
- 120 - 100 = 20
- 150 - 120 = 30
- 192 - 150 = 42
- 248 - 192 = 56
- 320 - 248 = 72
- 410 - 320 = 90

Now, the differences are:
- 20, 30, 42, 56, 72, 90

### Step 6: Calculate the second differences again
Now, let's calculate the second differences:
- 30 - 20 = 10
- 42 - 30 = 12
- 56 - 42 = 14
- 72 - 56 = 16
- 90 - 72 = 18

The second differences are:
- 10, 12, 14, 16, 18

### Conclusion
The differences are now consistent, indicating that the term 154 was indeed the wrong term. The correct term should be 150.

Thus, the wrong term in the series is **154**.

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