Find the wrong term in an incremental pattern: 125, 127, 130, 135, 142, 153, 165, … Exactly one number violates the intended rule.

Introduction / Context:
Many incremental series grow by adding consecutive prime numbers. Identifying such a pattern and checking each step quickly reveals an outlier. Here, small increments hint at +2, +3, +5, +7, etc., which are the first primes in order.

Given Data / Assumptions:

• Sequence: 125, 127, 130, 135, 142, 153, 165, …
• We expect consecutive-prime additions.
• Only one term should break the rule.

Concept / Approach:
Compute the differences between successive terms and compare them to the prime list: 2, 3, 5, 7, 11, 13, … If all but one step match, the number at which the mismatch occurs is the odd term.

Step-by-Step Solution:
127 − 125 = +2 (prime)130 − 127 = +3 (prime)135 − 130 = +5 (prime)142 − 135 = +7 (prime)153 − 142 = +11 (prime)165 − 153 = +12 (but the next prime should be +13)Therefore, the step to 165 is wrong; the correct next term would be 153 + 13 = 166.

Verification / Alternative check:
Replacing 165 with 166 restores perfect prime increments: +2, +3, +5, +7, +11, +13, which is a standard, widely used pattern.

Why Other Options Are Wrong:

• 130 / 142 / 153 each fits the prime-increment rule exactly.

Common Pitfalls:

• Assuming a mixed odd-number increment; primes are the clean fit and expose the 165 anomaly.

Final Answer:
165