Question Type: Number Series

The problem asks us to find the next number in the given sequence. This type of question tests the ability to identify the underlying logical pattern or rule governing the series.

Step-by-Step Solution:**

Let's analyze the given series to find the pattern. The series is: 4, 22, 43, 69, 102, 144, ?

Step 1: Find the difference between consecutive terms.

• 22 - 4 = 18
• 43 - 22 = 21
• 69 - 43 = 26
• 102 - 69 = 33
• 144 - 102 = 42

The series of differences is: 18, 21, 26, 33, 42.

Step 2: Check for a pattern in the new series of differences.

The differences are not constant, so let's find the difference of these differences (this is called a double difference or second-level difference).

• 21 - 18 = 3
• 26 - 21 = 5
• 33 - 26 = 7
• 42 - 33 = 9

The second-level difference series is: 3, 5, 7, 9.

Step 3: Identify the core logic/pattern.

The second-level differences (3, 5, 7, 9) form a simple arithmetic progression of consecutive odd numbers, with a common difference of 2.

Step 4: Predict the next terms based on the identified pattern.

• The next number in the second-level difference series (3, 5, 7, 9, ...) will be 9 + 2 = 11.
• Now, we add this value to the last term of the first-level difference series (18, 21, 26, 33, 42, ...). The next difference will be 42 + 11 = 53.
• Finally, we add this new difference to the last term of the original series to find the missing number.

Next Number = Last Term + Next Difference = 144 + 53 = 197.

Therefore, the number that will replace the question mark (?) is 197.

Elimination of Incorrect Options:

• Option 1 (177): If the next number were 177, the difference would be 177 - 144 = 33. This does not follow the pattern where the differences are increasing.
• Option 3 (179): If the next number were 179, the difference would be 179 - 144 = 35. This also breaks the logical progression.
• Option 4 (199): If the next number were 199, the difference would be 199 - 144 = 55. Our established pattern predicts the next difference to be 53 (42+11), not 55.
• Option 2 (197): This gives a difference of 197 - 144 = 53, which perfectly fits the double difference pattern we discovered.

Final Answer: The correct answer is 197.