In the number sequence 200, 200, 100, 300, 75, ?, identify the next term that should replace the question mark so that the underlying pattern is correctly maintained.

Introduction / Context:
This question presents a short numerical sequence with one missing term. The numbers appear to rise and fall, which often suggests alternating operations such as multiplying and dividing by different factors. Recognizing such alternating patterns is a common requirement in competitive examinations that test numerical and logical reasoning.

Given Data / Assumptions:

• The known sequence is: 200, 200, 100, 300, 75, ?
• There is a single missing term at the end, to be filled with one of the given options.
• We assume a single consistent pattern governs the entire sequence.
• Operations may alternate between multiplication and division, or between different constant factors.

Concept / Approach:
The first observation is that the values 200, 200, 100, 300, 75 fluctuate in a way that suggests ratios rather than simple differences. Comparing each term with the one immediately before it can reveal a pattern in terms of multipliers or divisors. If we discover a sequence of factors such as ×1, ÷2, ×3, ÷4, then we can naturally extend it to the next term by continuing that factor pattern (for example ×5).

Step-by-Step Solution:
Step 1: Compare each term with the previous one. 200 → 200: factor 200 / 200 = 1 (multiply by 1). 200 → 100: factor 100 / 200 = 1/2 (divide by 2). 100 → 300: factor 300 / 100 = 3 (multiply by 3). 300 → 75: factor 75 / 300 = 1/4 (divide by 4). Step 2: Recognize the pattern of factors: ×1, ÷2, ×3, ÷4. Step 3: This strongly suggests the next operation should be ×5 (continuing with 1, 2, 3, 4, 5 as multipliers or divisors). Step 4: Apply the factor ×5 to the last known term: 75 × 5 = 375. Step 5: Therefore the missing term is 375.

Verification / Alternative check:
We can list the pattern of operations explicitly: 200 × 1 = 200, 200 ÷ 2 = 100, 100 × 3 = 300, 300 ÷ 4 = 75, and now 75 × 5 = 375. The multipliers and divisors 1, 2, 3, 4, 5 are in natural ascending order, split alternately between multiplication and division. There is no inconsistency in this explanation, making the pattern clear and robust.

Why Other Options Are Wrong:
Options such as 150, 175 or 250 do not fit the systematic factor pattern. For instance, 75 × 2 = 150, but that would correspond to ×2, which breaks the smooth continuation after ÷4. Similarly, 175 and 250 require arbitrary factors that do not extend the observed sequence 1, 2, 3, 4. Thus they fail to maintain a simple, consistent rule across all terms.

Common Pitfalls:
A frequent mistake is to examine only the differences between consecutive terms (0, -100, +200, -225, …), which appear irregular and difficult to generalize. Another pitfall is to stop after testing only one or two possible rules instead of verifying the pattern across the entire sequence. Always ensure that any proposed rule works for every transition, not just part of the series.

Final Answer:
Continuing the alternating pattern of ×1, ÷2, ×3, ÷4 with ×5, the missing term must be 375.