In the number series 5, 26, 138, 717, 3649, the next term is _______. – UGC NET Question

The series can be modelled by a₁ = 5 and aₙ₊₁ = 5aₙ + n³ for n ≥ 1. Using this rule we obtain a₂ = 5·5 + 1³ = 26, a₃ = 5·26 + 2³ = 138, a₄ = 5·138 + 3³ = 717 and a₅ = 5·717 + 4³ = 3649, in agreement with the given numbers. For n = 5 the next term is a₆ = 5·3649 + 5³ = 18370. Hence 18370 is the only value that satisfies the same recurrence and properly extends the series.

Option A:
Option A, 18310, is 60 less than the value dictated by aₙ₊₁ = 5aₙ + n³ at n = 5. Choosing 18310 would require subtracting 60 from the correctly computed term, which has no basis in the earlier steps. Therefore this option does not conform to the established recurrence and is incorrect.

Option B:
Option B, 18340, is closer but still 30 less than the correct value 18370. It cannot be obtained from 5·3649 + 5³ and would imply weakening the cubic correction at the last stage. Since the recurrence is followed exactly before, option B is not a valid continuation.

Option C:
Option C, 18370, is exactly the result of applying the rule to a₅ with n = 5. It preserves the structure of multiplying the previous term by five and then adding the cube of the index. Because this process accounts for all earlier terms and produces 18370 next, this option is correct.

Option D:
Option D, 18430, is 60 greater than the computed value and again fails to satisfy 5a₅ + 5³. Accepting 18430 would mean amplifying the adjustment term at the last step without any justification. Thus option D breaks the recurrence and is not acceptable.