In the following number series, one term is incorrect: 1, 2, 6, 15, 31, 56, 91. Identify the wrong number that does not follow the pattern.

Introduction / Context:
Number series questions often hide a pattern in the differences between consecutive terms. Here, we must locate the single term that does not respect the underlying rule. Identifying such a term requires us to look at how each term grows from the previous one and to see whether there is a consistent formula that one term violates.

Given Data / Assumptions:

• Series: 1, 2, 6, 15, 31, 56, 91.
• Exactly one number in this list is wrong.
• We assume a reasonably simple pattern, often involving squares, cubes or polynomial type differences.

Concept / Approach:
A powerful method is to compute the differences between consecutive terms and then examine whether those differences follow a simpler pattern. If the differences themselves grow like 1, 4, 9, 16, and so on, then they may be consecutive squares. If one difference breaks that pattern, the term generating that difference is likely to be the mistake.

Step-by-Step Solution:
1. Write the given series: 1, 2, 6, 15, 31, 56, 91. 2. Compute first differences: 2 - 1 = 1 6 - 2 = 4 15 - 6 = 9 31 - 15 = 16 56 - 31 = 25 91 - 56 = 35 3. So the difference sequence is: 1, 4, 9, 16, 25, 35. 4. Notice that 1, 4, 9, 16, 25 are 1^2, 2^2, 3^2, 4^2, 5^2 respectively. 5. Therefore the next difference should be 6^2 = 36, not 35. 6. If the last difference were 36, then the last term should be 56 + 36 = 92. 7. The series should thus be 1, 2, 6, 15, 31, 56, 92 if it followed the perfect consecutive squares difference pattern. 8. Therefore 91 is the wrong term, since it breaks the squares pattern.

Verification / Alternative check:
Rebuild the series using the intended rule: Start from 1 and add 1^2 = 1 to get 2. Then add 2^2 = 4 to get 6. Add 3^2 = 9 to get 15. Add 4^2 = 16 to get 31. Add 5^2 = 25 to get 56. Add 6^2 = 36 to get 92. This confirms that a consistent pattern exists, and the term 91 does not fit.

Why Other Options Are Wrong:

• 31: If 31 were wrong, then differences around it would no longer follow consecutive squares. But 15 to 31 gives 16, which is 4^2 and fits the pattern.
• 15: The difference 15 - 6 = 9 is 3^2, so 15 is consistent.
• 56: The difference 56 - 31 = 25 is 5^2, so 56 also fits perfectly.

Common Pitfalls:
Many learners stop after noticing that the sequence increases roughly by larger and larger amounts, but they do not test if those increments are well known numbers like 1, 4, 9, and so on. Without computing all differences and checking for squares, it is easy to suspect the wrong term, especially in the middle of the sequence. Always check whether the differences form a known pattern such as consecutive squares or cubes.

Final Answer:
The wrong number in the series is 91.