In the series 311, 230, 294, ?, 281, 256, what number should replace the question mark?

Introduction / Context:
This question is a number series completion problem. The key skill required is to recognise a hidden pattern, often in the differences between consecutive terms, and then use that pattern to fill the missing value so that the sequence remains consistent.

Given Data / Assumptions:

• Series: 311, 230, 294, ?, 281, 256.
• Exactly one number is missing and must be determined.
• The sequence is assumed to follow a single logical pattern throughout.

Concept / Approach:
When the numbers do not show simple multiplication or division, the next strategy is to examine first-level differences. If these differences form a recognisable pattern, such as squares, cubes, or another arithmetic progression, we can leverage that to find the missing term.

Step-by-Step Solution:
Step 1: Find the difference between the first two terms: 311 − 230 = 81. Step 2: Difference between the second and third terms: 294 − 230 = 64. Step 3: Let the missing term be X. Then the difference from 294 to X and from X to 281 must fit a pattern. Check the later known pair: 281 − 256 = 25. Step 4: Observe the known differences: 81, 64, ?, ?, 25. Note that 81 = 9^2, 64 = 8^2, and 25 = 5^2. It suggests a sequence of decreasing perfect squares. Step 5: Continue the pattern of square differences: 9^2, 8^2, 7^2, 6^2, 5^2 which are 81, 64, 49, 36, 25. Step 6: From 294, subtract 49 (7^2) to get the missing term: X = 294 − 49 = 245. Step 7: Now check the next difference: 281 − 245 = 36, which is 6^2. This fits the square pattern perfectly.

Verification / Alternative check:
The completed series is 311, 230, 294, 245, 281, 256. The differences are 81, 64, 49, 36, 25 which are 9^2, 8^2, 7^2, 6^2, 5^2. The pattern of consecutive squares in decreasing order is clear and consistent for all terms, so 245 is uniquely correct.

Why Other Options Are Wrong:
If we insert 240, 244, or 246, the differences would not form a perfect square sequence. Either the square pattern breaks or one of the required square differences is lost. Only 245 maintains the ordered squares 9^2, 8^2, 7^2, 6^2, 5^2.

Common Pitfalls:
A common mistake is to look only at the numbers themselves rather than at their differences. Another error is to treat this as a random combination of operations, which leads to forced and inconsistent patterns. In exam series questions, perfect squares and cubes often appear in the differences, so you should actively check for them.

Final Answer:
The missing term that preserves the decreasing square difference pattern is 245.