Number series — alternating multipliers ×2 and ×3: Complete the series: 12, 24, 72, 144, 432, ?, ...

Introduction / Context:
This is a multiplication-based series where the pattern alternates between multiplying by 2 and by 3. Such alternating-multiplier patterns are common in aptitude tests and require checking consecutive term ratios.

Given Data / Assumptions:

• Series: 12, 24, 72, 144, 432, ?
• All terms are positive integers.
• Seek the unique next term that maintains the discovered rule.

Concept / Approach:
Compute successive ratios to test for consistent multipliers. If the multipliers follow a cycle, extend the cycle to the missing term.

Step-by-Step Solution:
24 = 12 * 272 = 24 * 3144 = 72 * 2432 = 144 * 3Next should be 432 * 2 = 864

Verification / Alternative check:
Continuing the cycle (*3, *2, ... ) would yield 864 * 3 = 2592 afterwards, confirming internal consistency.

Why Other Options Are Wrong:
728 and 852 do not respect the ×2 step; 1296 corresponds to ×3, which is not the next factor in the alternating pattern.

Common Pitfalls:
Assuming a pure geometric sequence with constant ratio; here the ratio alternates.

Final Answer:
864