In the number series 11, 19, 40, 87, 173, ?, one term is missing. By analyzing the pattern in the differences, determine the correct value for the missing term.

Introduction / Context:
This question involves a number series with differences that grow according to a secondary pattern. Instead of the terms themselves following a simple rule, the increments between them increase in a structured way. Recognizing patterns in differences and even differences of differences is an important reasoning skill.

Given Data / Assumptions:
The series given is:
11, 19, 40, 87, 173, ?
We assume that the differences between consecutive terms follow a progression that can be identified and used to compute the next term.

Concept / Approach:
First, we calculate the differences between consecutive terms. Then we inspect how these differences themselves change. If these second level changes follow a simple pattern, such as increasing by a constant amount, we can use that to extend the series consistently.

Step-by-Step Solution:
Step 1: Compute first differences between consecutive terms.19 - 11 = 840 - 19 = 2187 - 40 = 47173 - 87 = 86Step 2: List these first differences: 8, 21, 47, 86.Step 3: Compute second differences between these first differences.21 - 8 = 1347 - 21 = 2686 - 47 = 39Step 4: Notice that these second differences are multiples of 13 and increase by 13 each time: 13, 26, 39. The next second difference should therefore be 39 + 13 = 52.Step 5: The next first difference will be the last existing first difference plus this new second difference: 86 + 52 = 138.Step 6: Add this next first difference to the last term of the original series.173 + 138 = 311.

Verification / Alternative check:
We can check the extended difference pattern: starting with first differences 8, 21, 47, 86, and the new one 138. The second differences are 13, 26, 39, and 52, which are 13 multiplied by 1, 2, 3, and 4. This is a smooth and logical progression. Adding 138 to 173 to get 311 preserves this structure, confirming that 311 is the correct next term.

Why Other Options Are Wrong:
Values such as 301, 304, 294, or 320 would imply different first differences that do not create the clean series of second differences in steps of 13. For example, if the next term were 301, the last first difference would be 301 - 173 = 128, which does not extend the existing pattern of 8, 21, 47, and 86 in a way that fits the 13 based progression.

Common Pitfalls:
Some test takers focus only on the first differences and fail to see a pattern because those values themselves look irregular. Others may attempt to apply a linear or multiplicative rule on the main series, which does not work here. When first differences appear uneven but structured, always check second differences for a simple progression.

Final Answer:
The correct value that completes the number series is 311.