Find the wrong term in the number series 3, 4, 10, 32, 136, 685, 4116.

Introduction / Context:

This question asks for the identification of the incorrect term in a rapidly growing number series. The series is built using a compound rule of multiplication and addition, and one term has been deliberately altered. The candidate must reverse engineer the intended rule and see which term does not comply. Such problems are useful for assessing the ability to spot a single inconsistency in an otherwise well structured pattern.

Given Data / Assumptions:

• The given series is 3, 4, 10, 32, 136, 685, 4116.
• Exactly one term in this series is wrong.
• The correct series is generated by a uniform rule applied successively.
• The pattern likely involves multiplying by increasing integers and adding the same integer.

Concept / Approach:

A good starting point is to see whether each term can be produced from the previous one by multiplying with a certain integer and then adding that same integer. If this rule works for most transitions but fails at one point, the term at the failing point must be incorrect. In many exam series, factors increase by 1 each step, and the same increment is used for addition, leading to a rule of the type a(n+1) = a(n) * k + k.

Step-by-Step Solution:

Step 1: Test the rule from 3 to 4: 3 * 1 + 1 = 4; this fits the pattern multiply by 1 and add 1. Step 2: From 4 to 10: 4 * 2 + 2 = 8 + 2 = 10; now the factor and addition are both 2. Step 3: From 10 to 32: 10 * 3 + 3 = 33, but the series shows 32, so this term does not satisfy the rule. Step 4: From 10 using the intended rule, the correct next term should be 33, not 32. Step 5: Continue the correct series: 33 * 4 + 4 = 132 + 4 = 136, 136 * 5 + 5 = 680 + 5 = 685, 685 * 6 + 6 = 4110 + 6 = 4116, which all match the later terms.

Verification / Alternative check:

We can formalize the rule as a(n+1) = a(n) * (n) + (n) starting from a(1) = 3. Then a(2) = 3 * 1 + 1 = 4, a(3) = 4 * 2 + 2 = 10, a(4) = 10 * 3 + 3 = 33, a(5) = 33 * 4 + 4 = 136, a(6) = 136 * 5 + 5 = 685, a(7) = 685 * 6 + 6 = 4116. Every step except the one involving 32 is consistent with this formula. The presence of 32 instead of 33 at the fourth position is therefore the only deviation, naming 32 as the wrong term.

Why Other Options Are Wrong:

Term 136 is correct because it follows exactly from 33 under the rule multiply by 4 and add 4. Term 685 correctly results from 136 multiplied by 5 and then increased by 5. Term 4116 arises from 685 multiplied by 6 and increased by 6. Term 10 also correctly follows from 4. None of these violate the pattern. Only 32 cannot be obtained from 10 by multiplying by 3 and adding 3, so it alone is inconsistent with the intended rule.

Common Pitfalls:

Students often look at differences in such series, but the differences here are not simple and can be misleading. Others may notice only that numbers grow quickly and think the pattern is arbitrary. A further error is to assume that the wrong term must be near the end, whereas in many exam questions it is placed in the middle to force the candidate to check every transition. Carefully testing a consistent rule on each adjacent pair is the sure way to locate the incorrect term.

Final Answer:

The term that does not satisfy the rule a(n+1) = a(n) * n + n is 32, so 32 is the wrong term in the series.