In the number series 3, 11, 41, 141, 455, the next term is _______. – UGC NET Question

This sequence can be described recursively by a₁ = 3 and aₙ₊₁ = 3aₙ + 2n² for n ≥ 1. Using this rule we get a₂ = 3·3+2·1² = 11, a₃ = 3·11+2·2² = 41, a₄ = 3·41+2·3² = 141 and a₅ = 3·141+2·4² = 455, which matches the terms given. For n = 5 the next term is a₆ = 3·455+2·5² = 1365+50 = 1415. Thus 1415 is the only value that satisfies the same recurrence pattern for the next step.

Option A:
Option A, 1387, does not equal 3·455+2·5² and therefore fails to satisfy the recurrence relation. It would imply subtracting 28 from the required value without any change in the rule. Since the earlier terms fit the recurrence exactly, option A is inconsistent with the established pattern.

Option B:
Option B, 1415, is exactly the result of applying aₙ₊₁ = 3aₙ+2n² when n = 5. It respects the same combination of a multiple of the previous term and an index-dependent correction used in all earlier steps. Because the rule remains unchanged and still produces the next term, 1415 is the correct continuation of the series.

Option C:
Option C, 1431, overshoots the recurrence output by 16 and cannot be written as 3·455+2·5². It would require altering the coefficient or adding an extra constant at the last step, which is not supported by the data. Therefore option C is not a valid continuation.

Option D:
Option D, 1459, deviates even more from the computed value and likewise does not satisfy the stated recurrence. Using 1459 would break the close correspondence between the rule and the generated sequence, so option D is incorrect.