In the number series 6, 6, 12, 36, 144, 720, ?, each term is generated by a consistent multiplicative rule. What is the next number in the series?

Introduction / Context:
This is a multiplicative series where the numbers grow very quickly. Such sequences are often based on factorial like patterns or repeated multiplication by increasing integers. Identifying the exact multiplier at each step allows us to compute the next term without ambiguity.

Given Data / Assumptions:

• The series is: 6, 6, 12, 36, 144, 720, ?
• Only one term, the last, is missing.
• We assume the rule is multiplicative rather than additive, given the rapid growth.
• The multipliers may follow their own simple sequence, such as 1, 2, 3, 4, 5, 6.

Concept / Approach:
We examine the ratio between successive terms to detect how each is produced from the previous one. If these ratios form a simple progression, we can extend that progression and multiply the last known term by the correct factor to get the next term. This approach is particularly effective for series whose magnitudes grow quickly.

Step-by-Step Solution:
Step 1: Compute the ratios of consecutive terms. 6 (second term) divided by 6 (first term) = 1. 12 divided by 6 = 2. 36 divided by 12 = 3. 144 divided by 36 = 4. 720 divided by 144 = 5. Step 2: Recognise the pattern of multipliers: ×1, ×2, ×3, ×4, ×5. Step 3: The next multiplier in this natural sequence is ×6. Step 4: Apply this multiplier to the last known term: 720 × 6 = 4320. Step 5: Therefore, the missing term is 4320.

Verification / Alternative check:
We can reconstruct the series explicitly: start with 6, multiply by 1 to stay at 6, then multiply by 2 to get 12, multiply by 3 to get 36, multiply by 4 to get 144, and multiply by 5 to get 720. This confirms that the multipliers are indeed the natural numbers from 1 to 5. Continuing with a multiplier of 6 yields 4320, ensuring consistency of the pattern.

Why Other Options Are Wrong:
Numbers like 3547, 2154 or 1765 are not equal to 720 times a small integer and do not fit any obvious extension of the multiplier sequence. They would require arbitrary, non integral or irregular multipliers that are inconsistent with the simple pattern observed so far. Thus, they cannot be part of this logically constructed series.

Common Pitfalls:
One mistake is to look at differences, which are irregular and do not show a simple pattern here. Another is to miscalculate one of the ratios and therefore miss the straightforward sequence of multipliers. Carefully computing each ratio and checking that they form 1, 2, 3, 4, 5 prevents these errors and makes the series easy to extend.

Final Answer:
Following the pattern of multiplying by 1, 2, 3, 4, 5, the next term is obtained by multiplying 720 by 6, giving 4320.