In the sequence 3, 4, 14, 48, 576, 27648, which one number is the odd one out based on the rule that, from the third term onward, each term equals the product of the previous two?

Introduction / Context:
This question asks you to find the odd one out in the sequence 3, 4, 14, 48, 576, 27648. At first glance the numbers grow very quickly and the ratios between consecutive terms are not constant. Instead of looking for a simple geometric progression, we should test whether later terms can be generated as products of earlier ones.

Given Data / Assumptions:
The sequence is:

• 3
• 4
• 14
• 48
• 576
• 27648

We assume that there may be a rule connecting each term with the two immediately preceding terms, and that one of the given numbers breaks this rule.

Concept / Approach:
A powerful pattern to check is whether a term equals the product of the previous two terms. If we can find a consistent rule of the form a(n+1) = a(n) * a(n−1) for all but one term, that exceptional term is naturally identified as the odd one out. We will reconstruct a corrected sequence to see which number does not fit.

Step-by-Step Solution:
Step 1: Assume the first two terms are 3 and 4.Step 2: If each new term were the product of the previous two, then the third term should be 3 * 4 = 12.Step 3: Using this idea, generate further terms:Fourth term would be 4 * 12 = 48.Fifth term would be 12 * 48 = 576.Sixth term would be 48 * 576 = 27648.Step 4: This gives a consistent product based sequence: 3, 4, 12, 48, 576, 27648.Step 5: Compare this intended sequence with the given one: the only discrepancy is that the given third term is 14 instead of 12.

Verification / Alternative check:
Once we see that 3, 4, 12, 48, 576, 27648 follows the rule a(n+1) = a(n) * a(n−1), it is clear that 14 does not satisfy this rule because 3 * 4 = 12 and not 14. All other terms fit perfectly as products of their two predecessors. This confirms that 14 is the only term that breaks an otherwise very clean and elegant pattern.

Why Other Options Are Wrong:
The numbers 4, 48 and 27648 are essential components of the product pattern. Removing any of them would destroy the chain of products and make it impossible to maintain a consistent rule based on a(n+1) = a(n) * a(n−1). Only the value 14 can be changed to 12 without disturbing the rest of the sequence.

Common Pitfalls:
Students may first try to find a constant ratio or simple additive pattern, which does not work here. When ratios vary but terms grow dramatically, it is a good idea to test whether each term can be constructed from the previous two, often via multiplication. This perspective quickly exposes 14 as the misfit.

Final Answer:
The odd one out, which does not fit the product of previous two terms rule, is 14.