In the number sequence 1, 2, 6, 15, 31, 56, 91, which single term is incorrect because it does not follow the underlying numerical pattern?

Introduction / Context:
This question tests your ability to recognize patterns in a sequence of numbers and to identify which term breaks a clear logical rule. Such questions are very common in aptitude tests because they check both observation skills and understanding of number series.

Given Data / Assumptions:
The sequence given is 1, 2, 6, 15, 31, 56, 91. We are told that exactly one of these terms is wrong, and we must find the incorrect term based on a consistent numerical pattern in the rest of the sequence. We assume the pattern is simple and based on basic operations such as addition or multiplication.

Concept / Approach:
For such series, a good starting point is to calculate the difference between consecutive terms. Often, these differences themselves follow a simple pattern (such as squares, cubes, or arithmetic progression). Once a clear pattern is found, the term that does not fit that pattern is the wrong term.

Step-by-Step Solution:
Step 1: Write the sequence: 1, 2, 6, 15, 31, 56, 91.Step 2: Compute consecutive differences: 2 - 1 = 1, 6 - 2 = 4, 15 - 6 = 9, 31 - 15 = 16, 56 - 31 = 25, 91 - 56 = 35.Step 3: Observe the pattern in the differences: 1, 4, 9, 16, 25 are perfect squares (1^2, 2^2, 3^2, 4^2, 5^2). The next square should be 6^2 = 36.Step 4: According to this pattern, after 56, the next term should be 56 + 36 = 92, not 91.Step 5: Therefore, 91 is the only term that breaks the rule of adding consecutive perfect squares to get the next term.

Verification / Alternative check:
Start again from 1 and keep adding consecutive squares: 1 + 1 = 2, 2 + 4 = 6, 6 + 9 = 15, 15 + 16 = 31, 31 + 25 = 56, 56 + 36 = 92. This reconstructed series is 1, 2, 6, 15, 31, 56, 92, which matches all terms except the last one given in the question. This confirms that the pattern is consistent and 91 is indeed wrong.

Why Other Options Are Wrong:
31, 56 and 15 each fit perfectly into the pattern of adding consecutive squares. Removing any of them would break the square-difference rule in more than one place. Only replacing 91 by 92 restores a clean pattern across the entire sequence.

Common Pitfalls:
Students sometimes try to guess the wrong term by looking only at approximate growth or by assuming a multiplication pattern without checking differences. Another common mistake is to not compute all differences carefully, which can hide a tidy square pattern. Always check the full run of differences before deciding which term is wrong.

Final Answer:
The only term that does not follow the pattern of adding consecutive perfect squares is 91.