In the number series 4, 25, 157, 959, 5785, the next term is _______. – UGC NET Question

This sequence is defined by a₁ = 4 and aₙ₊₁ = 6aₙ + 2n² − 1 for n ≥ 1. Using this rule we get a₂ = 6·4 + 2·1² − 1 = 25, a₃ = 6·25 + 2·2² − 1 = 157, a₄ = 6·157 + 2·3² − 1 = 959 and a₅ = 6·959 + 2·4² − 1 = 5785, in agreement with the series. For n = 5 the next term is a₆ = 6·5785 + 2·5² − 1 = 34710 + 50 − 1 = 34759. Therefore 34759 is the only value that continues this recurrence pattern.

Option A:
Option A, 34735, is 24 less than the recurrence result and cannot be written as 6a₅ + 2·5² − 1. Accepting 34735 would mean lowering the correct computed term only at n = 5, which conflicts with the consistent use of the rule. Thus option A is not correct.

Option B:
Option B, 34747, is 12 smaller than the correct value 34759 and again does not satisfy the recurrence formula. It introduces a small but unjustified reduction in the last term. Hence option B fails to represent the actual pattern behind the sequence.

Option C:
Option C, 34759, exactly equals the value obtained from 6·5785 + 2·5² − 1. It maintains the structure of multiplying the previous term by six and adding a quadratic adjustment minus one. Because it fits all earlier steps and extends the same rule, this option is the correct answer.

Option D:
Option D, 34771, is 12 greater than the required value and likewise cannot be derived from the recurrence. Using 34771 would introduce an artificial increase at the final step that is not predicted by the formula. Therefore option D is not a valid continuation of the number series.