In each of the following questions a number serie is given. After the series a number is given followed by (a),(b),(c ),(d) and (e ). You have to complete the series starting with the number given, following the sequence of the original series and answer the questions that follow the series.
`{:(582,574,601,537,662,446),(204,(a),(b),(c),(d),(e)):}`
What will come in place of (d)?

Options

A. 284
B. 68
C. 174
D. 331

Answer

To solve the problem step by step, let's analyze the given series and find the required number for (d) in the new series starting from 204.

### Step 1: Analyze the Original Series
The original series is:
- 582, 574, 601, 537, 662, 446

Let's find the pattern in the original series:
1. From 582 to 574:
- 582 - 574 = 8 (decrease)
2. From 574 to 601:
- 601 - 574 = 27 (increase)
3. From 601 to 537:
- 601 - 537 = 64 (decrease)
4. From 537 to 662:
- 662 - 537 = 125 (increase)
5. From 662 to 446:
- 662 - 446 = 216 (decrease)

The differences are:
- Decrease by \(2^3 = 8\)
- Increase by \(3^3 = 27\)
- Decrease by \(4^3 = 64\)
- Increase by \(5^3 = 125\)
- Decrease by \(6^3 = 216\)

### Step 2: Identify the Pattern
The pattern alternates between decreasing and increasing, with the amount being the cube of consecutive integers:
- Decrease by \(2^3\)
- Increase by \(3^3\)
- Decrease by \(4^3\)
- Increase by \(5^3\)
- Decrease by \(6^3\)

### Step 3: Apply the Pattern to the New Series
Now we will apply this pattern to the new series starting from 204:
- Start with 204

1. **(a)**:
- Decrease by \(2^3 = 8\)
- \(204 - 8 = 196\)

2. **(b)**:
- Increase by \(3^3 = 27\)
- \(196 + 27 = 223\)

3. **(c)**:
- Decrease by \(4^3 = 64\)
- \(223 - 64 = 159\)

4. **(d)**:
- Increase by \(5^3 = 125\)
- \(159 + 125 = 284\)

### Conclusion
Thus, the value that will come in place of (d) is **284**.