What will come in place of the question mark in the following number series: 14, 18, 9, 25, 0, ?

Introduction / Context:
This series alternates between increases and decreases, and the magnitudes of these changes are perfect squares. The question tests whether you can observe both the sign of the change and the fact that the differences are 2^2, 3^2, 4^2, 5^2, and so on. Such alternating square difference patterns are a favourite type of number series in competitive exams.

Given Data / Assumptions:

• Series: 14, 18, 9, 25, 0, ?
• Exactly one term is missing at the end.
• The terms both increase and decrease, suggesting an alternating pattern.

Concept / Approach:
Compute the differences between consecutive terms and examine both their signs (positive or negative) and magnitudes. If you notice that the magnitudes are 4, 9, 16, 25, which are the squares of 2, 3, 4, 5, and the signs alternate plus, minus, plus, minus, then you can predict the next difference as the next square with the opposite sign. This approach makes it easy to find the missing term.

Step-by-Step Solution:
Step 1: Compute the differences between consecutive terms. 18 - 14 = +4. 9 - 18 = -9. 25 - 9 = +16. 0 - 25 = -25. Step 2: Observe the absolute values of these differences: 4, 9, 16, 25. Step 3: Recognise that 4 = 2^2, 9 = 3^2, 16 = 4^2, and 25 = 5^2. Step 4: Note the sign pattern of the differences: +, -, +, -. Step 5: Therefore, the next difference should follow the same rule: it should be positive (continuing the alternation) and have magnitude 6^2 = 36. Step 6: Apply this to the last known term: 0 + 36 = 36.

Verification / Alternative check:
Write the full extended series with the computed term: 14, 18, 9, 25, 0, 36. Differences are then +4, -9, +16, -25, +36, whose magnitudes are 2^2, 3^2, 4^2, 5^2, 6^2 and whose signs alternate +, -, +, -, +. This shows a perfectly regular pattern in both magnitude and sign, confirming that 36 is the only value that fits naturally at the end of the series.

Why Other Options Are Wrong:
If we choose 11, 23, or 20 instead, the last difference will either not be a perfect square or will not have the correct sign. For instance, 20 - 0 = 20, which is not equal to 36 and is not the square of an integer in this sequence. As a result, the series would break the observed rule of alternating differences with magnitudes 2^2, 3^2, 4^2, 5^2, 6^2. Therefore these options cannot be correct.

Common Pitfalls:
Some learners pay attention only to whether the numbers are increasing or decreasing and miss the structure in the magnitudes of the jumps. Others may see that 4, 9, 16 are squares but fail to notice that 25 is also a square and that the signs alternate. In questions where the series oscillates up and down, it is especially important to track both the absolute values and the signs of the differences.

Final Answer:
The correct term to fill the blank and continue the pattern is 36.