What should come in place of the question mark (?) in the following number series ?
(निम्नलिखित संख्या श्रृंखला में प्रश्नवाचक चिन्ह (?) के स्थान पर क्या आना चाहिए?)
`6" "9" "15" "27" "51" "?`

Options

A. 84
B. 99
C. 123
D. 75

Answer

To find the next number in the series 6, 9, 15, 27, 51, we will analyze the pattern of the differences between consecutive terms.

1. **Identify the series**:
The given series is: 6, 9, 15, 27, 51.

2. **Calculate the differences between consecutive terms**:
- From 6 to 9: \(9 - 6 = 3\)
- From 9 to 15: \(15 - 9 = 6\)
- From 15 to 27: \(27 - 15 = 12\)
- From 27 to 51: \(51 - 27 = 24\)

So, the differences are: 3, 6, 12, 24.

3. **Observe the pattern in the differences**:
Now, let's look at the differences:
- The difference from 3 to 6 is \(6 - 3 = 3\)
- The difference from 6 to 12 is \(12 - 6 = 6\)
- The difference from 12 to 24 is \(24 - 12 = 12\)

The differences between the differences are: 3, 6, 12.

4. **Identify the pattern in the second differences**:
The second differences also seem to follow a pattern:
- The difference from 3 to 6 is \(6 - 3 = 3\)
- The difference from 6 to 12 is \(12 - 6 = 6\)

It appears that each difference is being multiplied by 2:
- \(3 \times 2 = 6\)
- \(6 \times 2 = 12\)

5. **Calculate the next difference**:
Continuing this pattern, the next difference should be:
- \(12 \times 2 = 24\)

6. **Calculate the next term in the series**:
Now, we add this next difference to the last term in the series:
- \(51 + 24 = 75\)

7. **Final Calculation**:
The next term after 51 is 75.

Thus, the number that should come in place of the question mark (?) is **75**.