A number series is given with one term missing. Choose the correct alternative from the options that will complete the series: 30, 60, 360, 3600, ?

Introduction / Context:
This question features a rapidly increasing number series driven by multiplication. The multipliers themselves follow a pattern, and correctly identifying that pattern is the key to determining the missing term. Such problems test the ability to understand higher-order patterns in products rather than simple addition or subtraction.

Given Data / Assumptions:
Given series: 30, 60, 360, 3600, ?We must find the term that follows 3600.We assume that each term is obtained from the previous one by multiplying it with a systematically changing integer.

Concept / Approach:
When numbers grow very large very quickly, a multiplicative pattern is usually at work. The strategy is to compute the ratio between each pair of consecutive terms and check whether these ratios follow their own numeric pattern, often an arithmetic progression. Once that pattern is clear, we extend it to find the next multiplier and thus compute the missing term.

Step-by-Step Solution:
Step 1: Compute 60 / 30 = 2, so the second term is 30 multiplied by 2.Step 2: Compute 360 / 60 = 6, so the third term is 60 multiplied by 6.Step 3: Compute 3600 / 360 = 10, so the fourth term is 360 multiplied by 10.Step 4: The multipliers are 2, 6, 10, which form an arithmetic sequence with a common difference of 4.Step 5: The next multiplier should be 14, obtained by adding 4 to 10.Step 6: Multiply 3600 by 14: 3600 * 14 = 50400.

Verification / Alternative check:
Extend the series with the candidate term: 30, 60, 360, 3600, 50400. List the multipliers again: 30 * 2 = 60, 60 * 6 = 360, 360 * 10 = 3600, 3600 * 14 = 50400. The multipliers 2, 6, 10, 14 themselves form a clean arithmetic progression with difference 4. This confirms that 50400 follows naturally in the sequence and is the correct missing term.

Why Other Options Are Wrong:
Option 48500 does not equal 3600 multiplied by any member of the pattern 2, 6, 10, 14 and breaks the arithmetic progression in the multipliers.Option 50500 similarly fails to be 3600 times a multiplier that fits the established pattern, and introduces an inconsistent ratio.Option 40800 equals 3600 * 11.333..., which is not part of the integer multiplier sequence and does not align with the +4 step in the multipliers.

Common Pitfalls:
Some learners may mistakenly look for a constant multiplier, which is not present, or miscalculate products involving large numbers. Another common error is to confuse the pattern of the main terms with that of the multipliers. Focusing on the ratio between successive terms and then observing the progression of those ratios is the most reliable method here.

Final Answer:
The missing term that completes the multiplicative series is 50400.