In this verbal reasoning number series question, one term is missing. Carefully study the series 30, 62, 189, 760, ? and choose the correct alternative from the options that best completes the sequence.

Introduction / Context:
This is another number series question from verbal reasoning where one term is missing. The series given is 30, 62, 189, 760, ? and the task is to detect the underlying rule and use it to find the next term. Such questions are very popular in competitive exams because they test observation, logical thinking, and the ability to recognise numerical patterns quickly and accurately.

Given Data / Assumptions:
Given series: 30, 62, 189, 760, ?
Exactly one consistent pattern links each term to the next.
The same pattern must be applied step by step throughout the series.
Only one of the provided options will satisfy the pattern for the missing term.

Concept / Approach:
As with many non-trivial series, simple differences do not give a clear pattern here. Instead, we test operations where each term is generated from the previous term using multiplication combined with addition. If we attempt the pattern previous term * n + n (with n increasing gradually), we notice that it works perfectly: 30 * 2 + 2, 62 * 3 + 3, 189 * 4 + 4, and so on. The integers 2, 3, 4, 5 form a clear increasing sequence for the multipliers and addends.

Step-by-Step Solution:
From 30 to 62: 30 * 2 + 2 = 60 + 2 = 62. From 62 to 189: 62 * 3 + 3 = 186 + 3 = 189. From 189 to 760: 189 * 4 + 4 = 756 + 4 = 760. The pattern clearly uses multipliers 2, 3, and 4, each time adding the same integer as well. Next, we should use 5 as both multiplier and addend: 760 * 5 + 5 = 3800 + 5 = 3805. So, the missing term must be 3805.

Verification / Alternative check:
We verify by rechecking each transition. The formula previous term * n + n is valid for all known steps. The sequence of n as 2, 3, 4, and 5 is simple and logical. Substituting n = 5 produces 3805, which matches exactly one option. No other option maintains such a smooth progression, so the pattern is robust and not forced. This confirms that 3805 is the correct continuation of the series.

Why Other Options Are Wrong:
Option A: 3306 cannot be written as 760 * 5 + 5 or any simple extension of the earlier rule.
Option B: 1157 is far too small compared with the explosive growth of the series and does not satisfy the pattern.
Option C: 2185 also fails the multiplier plus addend logic observed earlier.
Option E: 1521 is unrelated to the established structure and would require a completely new and inconsistent rule.

Common Pitfalls:
Many candidates only check differences: 32, 127, 571, which look irregular and confusing. This may tempt them to guess. Another pitfall is assuming only one type of operation (pure multiplication or pure addition). In higher level exams, exam setters often combine multiplication with addition or subtraction, and the increment in the multiplier itself follows a simple sequence that must be noticed. Ignoring this can lead to wrong answers even when the pattern is actually very systematic.

Final Answer:
The missing number that correctly completes the series is 3805, so the correct option is 3805.