In the series 1, 2, 6, 15, 31, 56, x, choose the correct value of x so that the sequence continues to follow a consistent pattern.

Introduction / Context:
This is a classic increasing number series where each term is larger than the previous one by a growing amount. Such sequences often rely on patterns in the differences between consecutive terms, and those differences may themselves form squares, cubes or an arithmetic progression. The goal is to identify the structure and then use it to calculate the missing value x.

Given Data / Assumptions:

• The given series is: 1, 2, 6, 15, 31, 56, x.
• Only one term (x) is missing at the end.
• We assume a single simple rule, or a small set of related rules, generates all terms.
• Differences between consecutive terms are likely to hold the key.

Concept / Approach:
The simplest approach is to compute the first level differences between consecutive terms. If these differences follow a recognizable pattern (for example, they are perfect squares or form their own arithmetic series), we can extend that pattern to find the next difference. Adding that next difference to the last known term gives us x. This two layer structure is very common in reasoning tests: a series whose differences are themselves patterned.

Step-by-Step Solution:
Step 1: Compute the differences between consecutive terms. 2 - 1 = 1. 6 - 2 = 4. 15 - 6 = 9. 31 - 15 = 16. 56 - 31 = 25. Step 2: List the differences: 1, 4, 9, 16, 25. Step 3: Recognize that these are perfect squares: 1^2, 2^2, 3^2, 4^2, 5^2. Step 4: The next difference in this pattern should be 6^2 = 36. Step 5: Add this next difference to the last known term: 56 + 36 = 92. Step 6: Hence, x = 92.

Verification / Alternative check:
We can reconstruct the series from the beginning using the rule: start at 1, then keep adding consecutive squares: 1 + 1 = 2, 2 + 4 = 6, 6 + 9 = 15, 15 + 16 = 31, 31 + 25 = 56, and finally 56 + 36 = 92. Each step is consistent, and the pattern of squared differences is smooth and unbroken. This is strong evidence that the identified rule is correct.

Why Other Options Are Wrong:
Values like 65, 78 or 81 would require the next difference to be 9, 22 or 25 rather than 36, disrupting the pattern of perfect squares. Those values can only be obtained by using ad hoc differences rather than extending the simple 1^2, 2^2, 3^2, 4^2, 5^2, 6^2 pattern. Since such irregular changes are not supported by the earlier part of the series, these options are inconsistent and therefore incorrect.

Common Pitfalls:
Learners sometimes try to fit a direct formula to the nth term, which can be tedious and unnecessary here. Another mistake is to notice that the series appears irregular at first sight and stop exploring after checking only one or two differences. By carefully examining all differences and recognizing them as consecutive squares, the structure becomes simple and elegant.

Final Answer:
Continuing the pattern of adding consecutive squares to each term, the missing value x is 92.