In the following number series question, select the missing number that will correctly complete the series: 1728, 864, 432, 216, ?

Introduction / Context:
This question presents a decreasing number series. The task is to identify how each term is obtained from the previous one and then apply that rule to find the missing number. The values suggest a simple fractional or divisional relationship, which is a very common theme in quantitative aptitude tests.

Given Data / Assumptions:
Given series: 1728, 864, 432, 216, ?We must determine the next term after 216.We assume a consistent multiplicative or divisional factor between consecutive terms.

Concept / Approach:
Whenever a series consistently decreases and the numbers are halved or reduced proportionally, checking for division by a fixed factor is an efficient approach. Here, the numbers are all divisible by 2, and the sequence seems to become exactly half of the previous term each time. Confirming this pattern allows us to confidently compute the missing term using a simple division.

Step-by-Step Solution:
Step 1: Compute the ratio 1728 / 864 = 2, so 864 is half of 1728.Step 2: Compute the ratio 864 / 432 = 2, so 432 is half of 864.Step 3: Compute the ratio 432 / 216 = 2, so 216 is half of 432.Step 4: The pattern is clear: each term is the previous term divided by 2.Step 5: Apply the same pattern: 216 / 2 = 108.Step 6: Therefore, the missing term in the series is 108.

Verification / Alternative check:
Write the completed series: 1728, 864, 432, 216, 108. Check each step: 1728 / 2 = 864, 864 / 2 = 432, 432 / 2 = 216, and 216 / 2 = 108. The rule is perfectly consistent through all transitions. The halving pattern is simple, regular, and matches all known terms, confirming that 108 is the correct answer.

Why Other Options Are Wrong:
Option 54 would correspond to another halving step after 108, not immediately after 216, so it is one step too far in the sequence.Option 116 does not maintain the exact halving pattern, since 216 / 116 is not equal to 2 or any simple ratio matching earlier steps.Option 200 breaks the clear halving rule entirely, because 216 / 200 is not equal to 2 and does not follow any simple mathematical relationship observed earlier.

Common Pitfalls:
Some learners may try to apply subtraction instead of division, which fails here because subtracted differences would not stay consistent. Others may mistakenly jump two steps in the pattern and propose 54, which is indeed a term in the extended sequence but not the immediate next term. It is crucial to check each transition one by one and ensure the chosen answer corresponds to the correct position in the series.

Final Answer:
The missing number that completes the series is 108.