A number series is given, with one term missing. Choose the correct alternative from the given options that will complete the series: 5, 10, 30, ?, 600, 3600.

Introduction / Context:
This series 5, 10, 30, ?, 600, 3600 is strongly suggestive of repeated multiplication, because the later terms become very large. In such problems, the multipliers often follow a simple increasing pattern such as 2, 3, 4, 5, 6. Recognising these patterns helps solve many multiplicative series quickly.

Given Data / Assumptions:

• Series: 5, 10, 30, ?, 600, 3600.
• Options: 120, 100, 90, 60.
• We assume that each term can be obtained by multiplying the previous one by a specific integer.

Concept / Approach:
Check the ratios between known consecutive terms. If the ratios follow a simple integer pattern, then the missing term can be found by inserting the appropriate multiplication between the known numbers. Here, 10 / 5 = 2 and 30 / 10 = 3 suggest multipliers 2 and 3. We then attempt to continue this sequence of multipliers in a natural way.

Step-by-Step Solution:
Step 1: Compute ratios for known terms.10 / 5 = 2.30 / 10 = 3.Step 2: Predict the next multipliers.A natural pattern is to use consecutive integers as multipliers: 2, 3, 4, 5, 6.Step 3: Insert the missing term using multiplier 4.Missing term = 30 * 4 = 120.Step 4: Check the remaining terms with multipliers 5 and 6.120 * 5 = 600.600 * 6 = 3600.

Verification / Alternative check:
Write the completed sequence with its multipliers explicitly: 5 * 2 = 10, 10 * 3 = 30, 30 * 4 = 120, 120 * 5 = 600, 600 * 6 = 3600. The multipliers 2, 3, 4, 5, 6 form a clear and simple pattern of consecutive integers, and no step contradicts this. Hence, 120 is uniquely correct.

Why Other Options Are Wrong:
100: Would require a multiplier of 100 / 30, which is not an integer and does not fit the pattern of consecutive integer multipliers.90: Likewise gives 90 / 30 = 3, which would repeat the multiplier 3 and break the increasing multiplier pattern.60: Would make the next multiplier 600 / 60 = 10, which is inconsistent with the earlier ratios 2 and 3.

Common Pitfalls:
Sometimes, candidates try to apply addition or subtraction first because those operations are simpler. However, when numbers grow quickly, multiplication is usually the main driver. Always check simple ratios between terms in such cases. Once the pattern of multipliers is confirmed, the missing term becomes easy to compute.

Final Answer:
The missing term, obtained by multiplying 30 by 4, is 120.