A number series is given, with one term missing. Choose the correct alternative from the given options that will complete the series and identify the missing number in the sequence: 9, 21, 45, 93, ?, 381.

Introduction / Context:
This series 9, 21, 45, 93, ?, 381 is a fast growing sequence. It does not follow simple constant differences or ratios, but it does show a pattern in how the differences themselves change. Typically, such problems use doubling differences or similar exponential style growth to move from one term to the next.

Given Data / Assumptions:

• Series: 9, 21, 45, 93, ?, 381.
• Options: 189, 198, 188, 112.
• We assume that the pattern used so far continues consistently for the missing term.

Concept / Approach:
First, calculate the differences between consecutive terms. When differences themselves follow a clear pattern, such as doubling or any other simple relation, we can use that to determine the missing term. Here we will see that the differences 12, 24, and 48 double at each step, which strongly indicates that the next difference should also follow that doubling rule.

Step-by-Step Solution:
Step 1: Compute first differences.21 - 9 = 12.45 - 21 = 24.93 - 45 = 48.Step 2: Observe the pattern in the differences.The differences are 12, 24, 48. Each difference is double the previous one.Step 3: Extend this difference pattern.Next difference should be 48 * 2 = 96.Step 4: Find the missing term using this difference.Missing term = 93 + 96 = 189.Step 5: Confirm with the last given term.The next difference should be 96 * 2 = 192, and 189 + 192 = 381, which matches the last term given.

Verification / Alternative check:
With the full series 9, 21, 45, 93, 189, 381, the differences are 12, 24, 48, 96, 192. Each difference is exactly double the previous one from start to finish. There is no inconsistency, which confirms that 189 is the correct missing value.

Why Other Options Are Wrong:
198: Would give a difference of 105 from 93 to 198 and then 183 to 381, which do not follow the doubling pattern.188: Would produce differences that break the neat doubling sequence 12, 24, 48, 96, 192.112: Is far too small and would disrupt the steadily increasing differences.

Common Pitfalls:
One mistake is to search for a direct multiplicative relation between terms, such as multiplying by 2 and adjusting slightly, which can become messy. A more systematic method is to inspect the differences first. When differences show a simple and exact pattern, like doubling, that usually indicates the correct structure of the series.

Final Answer:
The missing number that preserves the doubling pattern of differences is 189.