In the number series 9, 11, 16, 26, ?, 69, what number should replace the question mark to continue the pattern correctly?

Introduction / Context:
This problem is a typical missing-term number series question, often asked in aptitude tests to assess pattern recognition and logical reasoning. The series is 9, 11, 16, 26, ?, 69, and we must identify the rule that governs how each term is formed from previous terms. Such questions encourage careful analysis of differences and second-level patterns in those differences.

Given Data / Assumptions:

• The given series is: 9, 11, 16, 26, ?, 69.
• We are to find the single missing number in place of the question mark.
• All terms are integers.
• The pattern is assumed to be consistent throughout the series.

Concept / Approach:
The usual method for such series is to compute first differences (subtract each term from the next) and look for a pattern. If the first differences themselves follow some rule, we use that rule to find the missing term. Sometimes second differences or other operations are required. Here, we start with first differences and look for a simple progression or pattern in them.

Step-by-Step Solution:
Step 1: Write down the series: 9, 11, 16, 26, ?, 69.Step 2: Compute the differences between consecutive terms where possible.Step 3: 11 - 9 = 2, 16 - 11 = 5, 26 - 16 = 10.Step 4: Let the missing term be x. Then the remaining difference is x - 26 and 69 - x.Step 5: Looking at 2, 5, and 10, we try to detect a pattern. Notice that 2, 5, 10 could be followed by 17 and 26, because the jumps between them are 3, 5, 7, 9, which form an increasing sequence of odd numbers.Step 6: Specifically, start with 2, then add 3 to get 5, add 5 to get 10, add 7 to get 17, and add 9 to get 26.Step 7: So the sequence of differences should be: 2, 5, 10, 17, 26.Step 8: We already have 2, 5, and 10. Now we assign x - 26 = 17 and 69 - x = 26.Step 9: From x - 26 = 17, we get x = 43.Step 10: Check the last difference: 69 - 43 = 26, which matches the expected pattern.

Verification / Alternative check:
Now that we have x = 43, write the full series: 9, 11, 16, 26, 43, 69. The differences are 2, 5, 10, 17, 26. Next, calculate the differences between these differences: 5 - 2 = 3, 10 - 5 = 5, 17 - 10 = 7, 26 - 17 = 9. These second differences form the sequence 3, 5, 7, 9, which are consecutive odd numbers. This confirms that the pattern is consistent and that x = 43 is the correct missing term.

Why Other Options Are Wrong:
Option 31: Produces differences 2, 5, 10, 5, 38 which show no consistent pattern.
Option 38: Gives differences 2, 5, 10, 12, 31; again, no regular structure in first or second differences.
Option 45: Leads to differences 2, 5, 10, 19, 24; the second differences do not form a simple sequence.
Option 41: Yields differences 2, 5, 10, 15, 28, which still breaks the neat pattern of second differences being consecutive odd numbers.

Common Pitfalls:
Students may stop after noticing a rough increase in differences and guess a value without checking higher-level patterns. Another common mistake is to try unrelated operations (such as multiplying or dividing terms) when a simpler difference-based pattern exists. It is usually best to start with first and second differences and only switch approaches if no reasonable pattern emerges.

Final Answer:
The missing number in the series is 43.