UGC NET Questions (Paper – 1)

The series can be modelled by the recurrence a₁ = 4 and aₙ₊₁ = 2aₙ + 3n² − 1 for n ≥ 1. Applying this rule gives a₂ = 2·4+3·1²−1 = 10, a₃ = 2·10+3·2²−1 = 31, a₄ = 2·31+3·3²−1 = 88 and a₅ = 2·88+3·4²−1 = 223, which matches the given sequence. For n = 5 the next term is a₆ = 2·223+3·5²−1 = 446+75−1 = 520. Therefore 520 is the unique next term that obeys this recurrence.

Option A:
Option A, 520, is exactly the value produced by the rule aₙ₊₁ = 2aₙ+3n²−1 when applied to a₅ = 223 with n = 5. It preserves the same structure of doubling the previous term and adding a square-based correction. Since all previous terms are generated in exactly this way, 520 is the correct continuation.

Option B:
Option B, 512, is slightly smaller than the recurrence output and does not satisfy 2·223+3·5²−1. It would require subtracting 8 from the correct value only at the final step. This arbitrary change is not reflected earlier in the sequence, so option B is not consistent with the pattern.

Option C:
Option C, 528, is larger than the correctly computed value and cannot be derived from the recurrence without altering its parameters. Choosing 528 would artificially inflate the sixth term and destroy the exact match between the rule and the data. Hence option C is incorrect.

Option D:
Option D, 536, deviates even more from 520 and again fails to arise from the relation aₙ₊₁ = 2aₙ+3n²−1. Using 536 would break the carefully defined index-dependent correction term, so this option is incorrect.

This series is described by the rule aₙ = n⁴ + 5n² for n starting from 1. Evaluating for n = 1, 2, 3, 4 and 5 yields 1+5 = 6, 16+20 = 36, 81+45 = 126, 256+80 = 336 and 625+125 = 750. For n = 6 we obtain 6⁴+5·6² = 1296+180 = 1476. Therefore 1476 is the only term that keeps this quartic-plus-quadratic pattern intact for the next position.

Option A:
Option A, 1440, is lower than the formula’s output and does not equal n⁴+5n² for n = 6. It would require subtracting 36 from the correctly computed term only at the sixth step, which is not implied by the earlier values. This inconsistency makes option A incorrect.

Option B:
Option B, 1452, still fails to match 1296+180 and thus does not follow the same expression. Although numerically close, it cannot be produced by the rule for the sixth term. Hence option B would break the algebraic structure of the series and is not acceptable.

Option C:
Option C, 1464, lies between some candidate values but again does not satisfy aₙ = n⁴+5n² when n = 6. Selecting 1464 would mean slightly reducing the correct result without any justification from the preceding pattern. Therefore option C is not logically consistent with the sequence.

Option D:
Option D, 1476, coincides exactly with the value computed from the rule for n = 6. It preserves both the powers and coefficients that define the behaviour of the earlier terms. Because this option emerges naturally from the same formula, 1476 is the correct continuation of the number series.