Analyzing the Number Sequence Pattern

We are given a sequence of numbers: 47, 11, 53, ?, 34, 75. We need to find the missing number represented by '?'. Let's analyze the relationship between the numbers in the sequence.

Step 1: Calculate the Sum of Digits for Each Number

A common pattern in such puzzles involves the sum of the digits of the numbers. Let's calculate this for the known numbers in the sequence:

• For 47: The sum of digits is $4 + 7 = 11$.
• For 11: The sum of digits is $1 + 1 = 2$.
• For 53: The sum of digits is $5 + 3 = 8$.
• For 34: The sum of digits is $3 + 4 = 7$.
• For 75: The sum of digits is $7 + 5 = 12$.

Let the missing number be represented by '$X$'. Let the sum of its digits be \(S_X\).

Step 2: Organize Sums in a Sequence

We can list the numbers and their corresponding sums of digits:

The sequence of sums is: 11, 2, 8, \(S_X\), 7, 12.

Step 3: Identify the Pattern in the Sums Sequence

Let's examine the sequence of sums: 11, 2, 8, \(S_X\), 7, 12. We can observe a pattern by looking at the sums at alternating positions:

• Sum of sums at odd positions (1st, 3rd, 5th): $11 + 8 + 7 = 26$.
• Sum of sums at even positions (2nd, 4th, 6th): \(2 + S_X + 12 = 14 + S_X\).

The pattern suggests that the sum of sums at odd positions should be equal to the sum of sums at even positions.

Step 4: Calculate the Missing Sum (\(S_X\))

Equating the sums from Step 3:

Solving for \(S_X\):

This means the sum of the digits of the missing number (?) must be 12.

Step 5: Find the Missing Number from Options

Now, let's check the sum of digits for each given option:

• Option 1: 41. Sum of digits = $4 + 1 = 5$.
• Option 2: 129. Sum of digits = $1 + 2 + 9 = 12$.
• Option 3: 29. Sum of digits = $2 + 9 = 11$.
• Option 4: 138. Sum of digits = $1 + 3 + 8 = 12$.

Both options 129 and 138 have a sum of digits equal to 12, satisfying the identified pattern. However, typically in such questions, there is a unique intended answer. Given the options, 129 fits the derived condition (\(S_X = 12\)).

Conclusion: The Missing Number

Based on the pattern derived from the sum of digits and the equality of alternating sums, the missing number should have digits that sum to 12. Option 129 fulfills this condition.