In the number series 11, 13, 30, 96, ?, what is the next term that should replace the question mark?

Introduction / Context:
This is a next term in series question where each new term is built using the previous term through a mixture of multiplication and addition. Recognising such composite operations is very important for numerical reasoning tests because they combine ideas of arithmetic progressions and geometric style growth.

Given Data / Assumptions:
- The series is 11, 13, 30, 96, ? .
- All terms are positive integers.
- There is a consistent rule involving the previous term to generate the next one.
- We need to choose the correct next term from the options.

Concept / Approach:
When numbers increase quickly, multiplication by increasing factors plus a small addition often explains the pattern. Here, we will try expressions of the form next term = current term * n + k, where n and k follow a simple rule linked to the position of the term. The observation is that multipliers 1, 2, 3, 4 combined with additions 2, 4, 6, 8 give a neat and consistent structure.

Step-by-Step Solution:
Step 1: Move from 11 to 13. 11 * 1 + 2 = 13. Step 2: Move from 13 to 30. 13 * 2 + 4 = 26 + 4 = 30. Step 3: Move from 30 to 96. 30 * 3 + 6 = 90 + 6 = 96. Step 4: Generalise the rule. We are multiplying by 1, 2, 3, 4 in order and adding 2, 4, 6, 8 respectively. Both the multiplier and the added value increase by 1 in each step. Step 5: Find the next term. From 96 to the next term: 96 * 4 + 8 = 384 + 8 = 392.

Verification / Alternative check:
Write the pattern clearly for each step:
From 11 to 13: multiply by 1, add 2.
From 13 to 30: multiply by 2, add 4.
From 30 to 96: multiply by 3, add 6.
From 96 to ? : multiply by 4, add 8.
Both components, multiplier and addition, form simple sequences, so the rule is very consistent. Substituting confirms that the only possible next term using this pattern is 392.

Why Other Options Are Wrong:
Option A (264) could be considered if we used only multiplication by 2.75 approximately, but that does not fit earlier steps neatly.
Option C (412) and Option D (459) do not arise from any simple extension of the (multiply by n, add 2n) pattern and therefore break the regular structure.
Option B (392) is the only value that preserves the clearly identified rule of multiplying by 4 and adding 8 after multiplying by 1, 2, and 3 and adding 2, 4, and 6.

Common Pitfalls:
A typical mistake is to look only at differences between terms, which here are 2, 17, 66 and do not follow a simple pattern at first glance. Students may give up or overcomplicate the series. A better approach is to check for multiplicative relationships and then notice regular additions. Always examine both multiplication and addition together when numbers grow more than linearly.

Final Answer:
The next term in the series is 392.