In the following verbal reasoning number series, one term is missing. Carefully observe the pattern in the series 15, 32, 99, 400, ? and choose the correct alternative from the given options that will complete the sequence.

Introduction / Context:
This question belongs to the verbal reasoning and number series category. The task is to recognise the hidden pattern in the given sequence 15, 32, 99, 400, ? and then select the correct missing term from the options. Such questions test a candidate's ability to observe numerical relationships and apply them consistently, which is very important for competitive exams and aptitude tests.

Given Data / Assumptions:
The given series is: 15, 32, 99, 400, ?
We assume that there is a single consistent rule that links each term to the next.
The next term must follow the same pattern as the previous steps.
Only one of the options should satisfy the discovered rule when extended to the missing position.

Concept / Approach:
For number series questions, a good starting point is to examine how each term is obtained from the previous one. Here, differences alone look irregular, so we try operations involving multiplication and addition together. The observation is that each term seems to be generated from the previous term by multiplying with an increasing integer and then adding the same integer. This produces a pattern of the form: next term = current term * n + n, where n increases step by step (2, 3, 4, 5, and so on).

Step-by-Step Solution:
From 15 to 32: 15 * 2 + 2 = 30 + 2 = 32. From 32 to 99: 32 * 3 + 3 = 96 + 3 = 99. From 99 to 400: 99 * 4 + 4 = 396 + 4 = 400. The multiplier and addend increase by 1 each time (2, 3, 4, ...), so the next step should use 5. From 400 to the missing term: 400 * 5 + 5 = 2000 + 5 = 2005. Thus, the missing number that continues the pattern correctly is 2005.

Verification / Alternative check:
We can verify the rule by checking every step. Each time we multiply the previous term by an integer n and then add the same integer n. The sequence of n values is 2, 3, 4, and then 5. The calculated terms exactly match the existing series up to 400, and extending the rule gives 2005. No other option, when inserted, preserves such a smooth and consistent pattern. This confirms that the logic is correct and that 2005 is the only suitable continuation.

Why Other Options Are Wrong:
Option A: 2001 does not fit the rule current term * 5 + 5, since 400 * 5 + 5 is not 2001.
Option B: 2004 is also inconsistent with the same pattern and breaks the multiplier plus addend logic.
Option D: 1994 similarly fails the rule and would require a completely different and irregular pattern.
Option E: 2010 does not come from any simple extension of the established operation and is therefore not valid.

Common Pitfalls:
A common mistake is to focus only on the differences between terms (17, 67, 301) and assume that the pattern is too irregular, causing candidates to guess. Another pitfall is to search only for simple arithmetic or geometric progressions instead of exploring combined operations like multiply then add. In many exam questions, the key is to notice that both the multiplier and the number added increase in a regular fashion, which is easy to miss under time pressure.

Final Answer:
The number that correctly completes the series is 2005, so the correct option is 2005.