10, 34,74, 130, 204, 290

Options

A. 34
B. 74
C. 130
D. 204

Explanation

To solve the series: 10, 34, 74, 130, 204, 290, we need to identify the pattern in the sequence.

### Step 1: Identify the Pattern
Let's look at the differences between the consecutive terms in the series.

- The first term is 10.
- The second term is 34.
- The third term is 74.
- The fourth term is 130.
- The fifth term is 204.
- The sixth term is 290.

### Step 2: Calculate the Differences
Now, let's calculate the differences between each pair of consecutive terms:

1. 34 - 10 = 24
2. 74 - 34 = 40
3. 130 - 74 = 56
4. 204 - 130 = 74
5. 290 - 204 = 86

So, the differences are: 24, 40, 56, 74, 86.

### Step 3: Calculate the Differences of Differences
Next, let's calculate the differences of these differences:

1. 40 - 24 = 16
2. 56 - 40 = 16
3. 74 - 56 = 18
4. 86 - 74 = 12

So, the second differences are: 16, 16, 18, 12.

### Step 4: Identify the Pattern in Second Differences
The second differences do not form a consistent pattern, so let's analyze the original terms more closely.

### Step 5: Check for a Formula
We can see that the series might be generated by summing squares of odd numbers. Let's check:

1. 1^2 = 1
2. 3^2 = 9
3. 5^2 = 25
4. 7^2 = 49
5. 9^2 = 81
6. 11^2 = 121
7. 13^2 = 169

Now, let's see how these squares can be combined:

- 1^2 + 3^2 = 1 + 9 = 10
- 3^2 + 5^2 = 9 + 25 = 34
- 5^2 + 7^2 = 25 + 49 = 74
- 7^2 + 9^2 = 49 + 81 = 130
- 9^2 + 11^2 = 81 + 121 = 202 (not 204)
- 11^2 + 13^2 = 121 + 169 = 290

### Step 6: Identify the Incorrect Term
From our calculations, we see that instead of 204, the correct term should be 202. Therefore, the incorrect term in the series is 204.

### Conclusion
The number that does not fit the pattern in the series is **204**.

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