Number series – continue the cube-plus pattern: 2, 9, 28, 65, 126, … What is the next term?

Introduction / Context:Many competition series come from simple algebraic forms (squares or cubes) with small adjustments. Here, the values align with n^3 + 1.

Observation / Approach:

• 1^3 + 1 = 2
• 2^3 + 1 = 9
• 3^3 + 1 = 28
• 4^3 + 1 = 65
• 5^3 + 1 = 126

Step-by-Step Solution:Next uses n = 6: 6^3 + 1 = 216 + 1 = 217.

Verification / Alternative check:Differences grow as 7, 19, 37, 61, which match the expected cubic growth when adding 1 to n^3.

Why Other Options Are Wrong:195, 199, 205, 208 do not equal 6^3 + 1; they would break the clean cubic rule.

Common Pitfalls:Trying to fit differences to ad-hoc multipliers rather than spotting the direct polynomial pattern.

Final Answer:217