In the series of numbers 125, 127, 130, 135, 142, 153, 165, which term is the odd one out because it breaks the pattern of increasing differences based on prime numbers?

Introduction / Context:
This problem presents a number sequence whose terms increase according to a pattern involving prime numbers. The task is to identify which term does not conform to this pattern. Such questions are designed to assess your ability to link series differences with known special numbers, such as primes.

Given Data / Assumptions:
The sequence is: 125, 127, 130, 135, 142, 153, 165. We assume that the link between consecutive terms is based on adding successive prime numbers (for example 2, 3, 5, 7, 11, 13, and so on). One term is incorrect.

Concept / Approach:
The first step is to compute the differences between consecutive terms and check whether those differences follow a familiar pattern. Because the increments are small and seem to grow irregularly, it is natural to test whether they match the early prime numbers in order: 2, 3, 5, 7, 11, 13, etc.

Step-by-Step Solution:
Step 1: Compute the differences between consecutive terms.127 - 125 = 2.130 - 127 = 3.135 - 130 = 5.142 - 135 = 7.153 - 142 = 11.165 - 153 = 12.Step 2: Now observe the differences: 2, 3, 5, 7, 11 are all prime numbers in increasing order.Step 3: Following this pattern, after adding 11, the next logical difference should be the next prime, which is 13, not 12.Step 4: If we add 13 to 153, we get 153 + 13 = 166. So the term after 153 should be 166 to maintain the prime-difference pattern.Step 5: The given value is 165 instead of 166. Therefore, 165 is the unique term that does not fit the established pattern.

Verification / Alternative check:
Rewriting the sequence using the intended pattern, we have: 125 + 2 = 127, 127 + 3 = 130, 130 + 5 = 135, 135 + 7 = 142, 142 + 11 = 153, and 153 + 13 = 166. In this corrected series, all increments match consecutive primes. The only discrepancy between the given and the corrected series is at the last term, confirming that 165 is the odd one out.

Why Other Options Are Wrong:
127, 142 and 153 all help maintain the prime differences 2, 3, 5, 7 and 11. If you remove any of these terms, you destroy the consistent sequence of prime increments. Only changing the last term allows the full pattern with consecutive primes to remain intact.

Common Pitfalls:
Some students may suspect a more complex polynomial relationship, or may think of squares or cubes. Others may not recognize the list of differences as prime numbers and thus miss the key observation. Always check whether differences correspond to familiar special sequences such as primes, odd numbers or square numbers.

Final Answer:
The only term that breaks the pattern of adding consecutive primes is 165.