In the number series 3, 4, 10, 33, 136, what number should replace the question mark as the next term in the sequence?

Introduction / Context:
This question is a non standard number series problem designed to test the ability to identify a pattern where each term depends on the previous term multiplied by an increasing integer and then adjusted by the same integer. Recognizing such mixed operations is a core skill in number series questions used in many competitive exams.

Given Data / Assumptions:
- The series given is 3, 4, 10, 33, 136, ?. - We assume there is a unique consistent rule linking each term to the previous term. - We want the next term that maintains that rule.

Concept / Approach:
- Look for relationships of the form next term = current term * k + k, where k changes in a regular way. - Compare how the coefficient and constant term might depend on the position in the series. - Verify the pattern across all known transitions before extending it.

Step-by-Step Solution:
Step 1: From 3 to 4, we can write 3 * 1 + 1 = 4. Step 2: From 4 to 10, we can write 4 * 2 + 2 = 10. Step 3: From 10 to 33, we can write 10 * 3 + 3 = 33. Step 4: From 33 to 136, we can write 33 * 4 + 4 = 136. Step 5: The pattern is now clear: next term = current term * n + n, where n is 1, 2, 3, 4, and so on. Step 6: For the next transition, the multiplier and the addend should both be 5, so we compute 136 * 5 + 5. Step 7: 136 * 5 = 680, and 680 + 5 = 685.

Verification / Alternative check:
We can summarise the recurrence as a_(k+1) = a_k * (k) + (k) where k starts at 1 for the first step. Checking each step: - 3 to 4 uses k = 1. - 4 to 10 uses k = 2. - 10 to 33 uses k = 3. - 33 to 136 uses k = 4. The pattern holds for all known steps, so using k = 5 for the next term is logically consistent and yields 685.

Why Other Options Are Wrong:
Option A (150): This would require 136 * k + k = 150 for some single integer k, which is not an integer solution consistent with the earlier pattern. Option B (298): Does not fit the rule 136 * 5 + 5 and would break the linear increase of the coefficient. Option C (463): Cannot be expressed as 136 * 5 + 5 or 136 * 5 + any simple k consistent with earlier steps. Option D (572): Looks like 136 * 4 + 28, which does not match the clean rule of multiplying and then adding the same integer.

Common Pitfalls:
- Looking only at differences between terms instead of considering multiplicative patterns combined with additions. - Assuming the pattern is based on fixed multiplication or fixed addition, which fails in this series. - Not verifying the guessed pattern on every step, which can lead to selecting a number that coincidentally fits one gap but not the entire series.

Final Answer:
The correct next term that follows the pattern a_(k+1) = a_k * k + k is 685.