What should come in place of the question mark in the following number series: 1, 3, 24, 360, 8640, 302400, ?

Introduction / Context:
This question tests your understanding of multiplicative number series where each term is obtained by multiplying the previous term with a changing factor. To solve it correctly, we must carefully identify the exact pattern in the multipliers and then extend that pattern to find the missing term.

Given Data / Assumptions:
- The given series is: 1, 3, 24, 360, 8640, 302400, ?- All terms are positive integers that grow rapidly.- The pattern is expected to be systematic and consistent for every step in the series.

Concept / Approach:
The natural way to approach this series is to look for a relation between consecutive terms using multiplication. Once we compute the ratios of consecutive terms, we can try to recognise a secondary pattern in those multipliers, such as squares, products, or a simple sequence. After identifying the pattern of multipliers, we use it to compute the next multiplier and hence the missing term.

Step-by-Step Solution:
- From 1 to 3: multiplier = 3 / 1 = 3.- From 3 to 24: multiplier = 24 / 3 = 8.- From 24 to 360: multiplier = 360 / 24 = 15.- From 360 to 8640: multiplier = 8640 / 360 = 24.- From 8640 to 302400: multiplier = 302400 / 8640 = 35.- So the sequence of multipliers is: 3, 8, 15, 24, 35.- Now find the pattern in multipliers: differences are 5, 7, 9, 11 which increase by 2 each time.- Thus the next difference should be 13, so the next multiplier = 35 + 13 = 48.- Therefore the missing term = 302400 * 48 = 14515200.

Verification / Alternative check:
- Recreate the multipliers: 3, 8, 15, 24, 35, 48.- Check the differences again: 5, 7, 9, 11, 13 which form an arithmetic progression with common difference 2.- This confirms the internal consistency of the pattern and validates the computed term 14515200.

Why Other Options Are Wrong:
- 14525100: Very close numerically but does not match 302400 * 48 exactly, so it breaks the exact multiplicative rule.- 154152000: This is far larger than 302400 * 48 and would require a different, unjustified multiplier.- 15425100: Again, this number cannot be expressed as 302400 multiplied by an integer that fits the multiplier pattern.- Only 14515200 is consistent with both the term-to-term multiplication and the pattern in the multipliers.

Common Pitfalls:
Candidates often try to guess the answer by approximate growth or rough mental multiplication, which leads to close but incorrect values. Another common error is to look only at the original terms and ignore the simpler structure hidden in the sequence of multipliers. It is also easy to miscalculate one of the products and then carry that error forward. Always compute the ratios carefully and double-check the multiplication for the final step.

Final Answer:
The next term in the series is 14515200, so the correct option is 14515200.