What will come in place of question mark (? ) in the following number series ?
`8 " " 9 " " 22 " " 75 " " ? " " 1605 `

Options

A. 316
B. 318
C. 306
D. 320

Answer

To find the missing number in the series `8, 9, 22, 75, ?, 1605`, we can analyze the pattern in the sequence.

### Step-by-Step Solution:

1. **Identify the Pattern**:
- The given numbers are: 8, 9, 22, 75, ?, 1605.
- Let's look at how each number relates to the next one.

2. **Examine the First Transition (8 to 9)**:
- From 8 to 9:
- \( 8 \times 1 + 1 = 9 \)

3. **Examine the Second Transition (9 to 22)**:
- From 9 to 22:
- \( 9 \times 2 + 4 = 22 \)

4. **Examine the Third Transition (22 to 75)**:
- From 22 to 75:
- \( 22 \times 3 + 9 = 75 \)

5. **Identify the Next Transition (75 to ?)**
- We can see a pattern where each number is multiplied by an increasing integer (1, 2, 3, ...) and then added to the square of that integer:
- For 75, we will multiply by 4 and add \(4^2\) (which is 16):
- \( 75 \times 4 + 16 = 300 + 16 = 316 \)

6. **Verify the Next Transition (316 to 1605)**:
- Now, we need to check if this pattern holds for the next number:
- \( 316 \times 5 + 25 = 1580 + 25 = 1605 \)

7. **Conclusion**:
- The missing number in the series is **316**.

### Final Answer:
The number that comes in place of the question mark is **316**.