UGC NET Questions (Paper – 1)

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.

The pattern can be written as a₁ = 5 and aₙ₊₁ = 4aₙ + n³ + 2 for n ≥ 1. Applying this rule gives a₂ = 4·5 + 1³ + 2 = 23, a₃ = 4·23 + 2³ + 2 = 102, a₄ = 4·102 + 3³ + 2 = 437 and a₅ = 4·437 + 4³ + 2 = 1814, all of which match the series. For n = 5 the next term is a₆ = 4·1814 + 5³ + 2 = 7256 + 125 + 2 = 7383. Thus 7383 is the only value that satisfies this recurrence and correctly continues the sequence.

Option A:
Option A, 7359, is 24 less than the recurrence output for n = 5 and cannot be written as 4a₅ + 5³ + 2. Accepting 7359 would require subtracting 24 from the correctly computed term only at the final step. This violates the consistency of the recurrence, so option A is incorrect.

Option B:
Option B, 7371, is 12 smaller than the correct value 7383 and likewise does not equal 4·1814 + 125 + 2. It provides an approximate but not exact continuation and therefore does not follow the specified rule. Consequently, option B cannot be chosen as the next term.

Option C:
Option C, 7395, is 12 greater than the correct value and again fails to match the recurrence formula. To obtain 7395 we would have to add extra units to the computed result at n = 5, altering the defined update pattern. Hence option C is not a valid extension of the sequence.

Option D:
Option D, 7383, exactly equals the value produced by the rule aₙ₊₁ = 4aₙ + n³ + 2 when n = 5. It preserves the combination of quadrupling the previous term with an added cubic and constant adjustment. Because this structure explains every step of the series, 7383 is the correct answer.