A number series is given with one term missing. Select the correct alternative from the given options that will complete the pattern and determine the missing number in the sequence: 4, 13, 40, 121, ?

Introduction / Context:
This question involves a non linear number series where each term is generated from the previous term using a consistent rule. The sequence 4, 13, 40, 121, ? increases rapidly, which suggests a combination of multiplication and addition rather than simple constant differences. Such patterns are very common in reasoning and aptitude exams.

Given Data / Assumptions:

• Series: 4, 13, 40, 121, ?
• Options: 350, 360, 364, 362.
• We assume one fixed rule applies between every pair of consecutive terms.

Concept / Approach:
For fast growing series, it is natural to test multiplicative patterns such as "multiply by a constant and add or subtract a small number." If differences themselves grow in a nearly multiplicative way, that reinforces this idea. Here, testing the relation T(n+1) = 3 * T(n) + 1 quickly shows a perfect fit for the first three transitions, which can then be extended to find the missing term.

Step-by-Step Solution:
Step 1: Check relation from 4 to 13.3 * 4 + 1 = 12 + 1 = 13.Step 2: Check relation from 13 to 40.3 * 13 + 1 = 39 + 1 = 40.Step 3: Check relation from 40 to 121.3 * 40 + 1 = 120 + 1 = 121.Step 4: Apply the same pattern to 121.Next term = 3 * 121 + 1 = 363 + 1 = 364.

Verification / Alternative check:
Write the complete series using the found term: 4, 13, 40, 121, 364. Now verify every jump: 4 to 13, 13 to 40, 40 to 121, and 121 to 364 all follow the exact rule "multiply by 3 and then add 1." There is no break or inconsistency anywhere, which confirms that 364 fits perfectly. The rapid growth and uniform transformation at each step support this as the unique valid answer.

Why Other Options Are Wrong:
350: This cannot be written as 3 * 121 + constant with the same constant as previous steps.360: Also does not satisfy T(n+1) = 3 * T(n) + 1 for n = 4th term.362: Fails the established relation and breaks the pattern.

Common Pitfalls:
Some candidates may look only at differences: 9, 27, 81 and guess that the next difference is 243, which coincidentally leads to the same answer 364. However, they may not articulate the underlying relation clearly. It is safer to express the core pattern directly as T(n+1) = 3 * T(n) + 1 so that the logic remains easy to verify and apply to similar problems.

Final Answer:
The missing term that follows the rule T(n+1) = 3 * T(n) + 1 is 364.