What number should replace the question mark in the series 3240, 540, 108, 27, ?, 4.5 so that the pattern is maintained?

Introduction:
This question is a number series puzzle where each term is obtained from the previous term by applying a specific operation. Recognizing the pattern, particularly when it involves division by decreasing integers, is key to finding the missing term.

Given Data / Assumptions:

• The given series is: 3240, 540, 108, 27, ?, 4.5.
• Exactly one term is missing, marked by a question mark.
• We must deduce the rule connecting consecutive terms and then apply it to find the missing number.

Concept / Approach:
We compare each term with the previous one to see whether multiplication, division or a combination is being used. Here, the pattern turns out to be division by successively smaller integers: 6, 5, 4, 3, and 2.

Step-by-Step Solution:
Step 1: Compare 3240 and 540.3240 ÷ 540 = 6, so 540 = 3240 ÷ 6.Step 2: Compare 540 and 108.540 ÷ 108 = 5, so 108 = 540 ÷ 5.Step 3: Compare 108 and 27.108 ÷ 27 = 4, so 27 = 108 ÷ 4.Step 4: Continue the same logic.To go from 27 to the next term, divide by 3.Next term = 27 ÷ 3 = 9.Step 5: Confirm the last step.Then 9 ÷ 4.5 = 2, so 4.5 = 9 ÷ 2, consistent with dividing by 2 at the final step.

Verification / Alternative check:
The divisor sequence is 6, 5, 4, 3, 2, which is a simple descending pattern. Each division yields the next term correctly, and the missing value 9 fits perfectly into this scheme, maintaining the structure of the series.

Why Other Options Are Wrong:
Values like 7, 8, 10 or 6 would break the clean pattern of division by 6, 5, 4, 3 and 2. Only 9 ensures that the ratio between consecutive terms remains consistent with this rule.

Common Pitfalls:
Some candidates may look for addition or subtraction patterns instead of multiplicative ones. Others stop after checking only one or two steps and fail to notice the full descending divisor sequence.

Final Answer:
The missing number in the series is 9.