Number series (n^3 − 1 pattern): 7, 26, 63, 124, 215, 342, (…) Identify the next term generated by the cubic-minus-one rule.

Introduction / Context:Some sequences are built from polynomial expressions of n. Here the terms closely match cubes minus one, a common pattern in competitive quantitative questions.

Given Data / Assumptions:

• Observed terms: 7, 26, 63, 124, 215, 342.
• Test whether each term equals k^3 − 1 for successive integers k.

Concept / Approach:Compute small cubes and subtract 1: 2^3 − 1 = 7, 3^3 − 1 = 26, 4^3 − 1 = 63, and so on. If this fits all given terms, continue with the next integer.

Step-by-Step Solution:

Verification / Alternative check:Confirm monotonic growth matching cubic trend; differences rise consistently, as expected for polynomial sequences.

Why Other Options Are Wrong:

• 481, 391, 421: Do not equal any (n^3 − 1) for n ∈ N after 7.

Common Pitfalls:Assuming arithmetic or geometric progressions; always test simple power-based constructions (squares, cubes) when numbers align closely with known powers.

Final Answer:511