Number series — find the next term using second-difference logic. Sequence: 7, 26, 63, 124, 215, 342, ?

Introduction / Context: Many non-linear sequences grow via increasing differences; the differences themselves follow a clear pattern. Detecting that the second differences are in arithmetic progression allows you to project the next first difference and hence the next term.

Given Data / Assumptions:

• Sequence: 7, 26, 63, 124, 215, 342, ?
• We examine first differences and then second differences.

Concept / Approach: Compute first differences: 26−7=19, 63−26=37, 124−63=61, 215−124=91, 342−215=127. Now compute second differences between these: 37−19=18, 61−37=24, 91−61=30, 127−91=36. These second differences increase by +6 each time (18, 24, 30, 36, next 42).

Step-by-Step Solution: First differences so far: 19, 37, 61, 91, 127. Second differences: +18, +24, +30, +36 (step +6). Next second difference = 36 + 6 = 42. Next first difference = last first difference + next second difference = 127 + 42 = 169. Next term = last term + next first difference = 342 + 169 = 511.

Verification / Alternative check: 169 is 13^2, which often appears in such crafted puzzles; adding it to 342 produces 511, a value consistent with the accelerating growth observed.

Why Other Options Are Wrong: 391/421/481: These would imply different first differences that break the +6 pattern in second differences.

Common Pitfalls: Trying to match powers or products directly without checking the structured growth of differences.

Final Answer: 511