What should come in place of question mark (?) in the following number series ?
`"9 73 198 ? 757 1269"`

Options

A. A) 396
B. B) 594
C. C) 404
D. D) 414

Answer

To find the missing number in the series `9, 73, 198, ?, 757, 1269`, we will analyze the differences between the consecutive numbers.

### Step 1: Calculate the differences between the given numbers.

1. **First difference**:
- \( 73 - 9 = 64 \)

2. **Second difference**:
- \( 198 - 73 = 125 \)

3. **Third difference**:
- We don't know this yet, as it corresponds to the missing number.

4. **Fourth difference**:
- \( 757 - ? \) (we'll calculate this later)

5. **Fifth difference**:
- \( 1269 - 757 = 512 \)

### Step 2: Identify the pattern in the differences.

Now, let's summarize the differences we have calculated so far:
- The first difference is \( 64 \).
- The second difference is \( 125 \).
- The fifth difference is \( 512 \).

### Step 3: Recognize the cubes in the differences.

We notice that:
- \( 64 = 4^3 \)
- \( 125 = 5^3 \)
- \( 512 = 8^3 \)

### Step 4: Determine the missing difference.

Following the pattern of cubes, we can see that:
- The first difference corresponds to \( 4^3 \)
- The second difference corresponds to \( 5^3 \)
- The missing difference should correspond to \( 6^3 = 216 \).

### Step 5: Calculate the missing number.

Now we can find the missing number by adding the missing difference \( 216 \) to the last known number \( 198 \):
- \( ? = 198 + 216 = 414 \)

### Step 6: Verify the next difference.

To ensure our solution is correct, we can check the next difference:
- Adding \( 343 \) (which is \( 7^3 \)) to \( 414 \):
- \( 414 + 343 = 757 \) (which matches the next number in the series).

### Conclusion

Thus, the missing number in the series is **414**.