UGC NET Questions (Paper – 1)

Here the pattern is aₙ = n⁴ + 2n³ + 1 with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1+2+1 = 4, 16+16+1 = 33, 81+54+1 = 136, 256+128+1 = 385 and 625+250+1 = 876, which matches the sequence exactly. For n = 6 we compute 6⁴+2·6³+1 = 1296+432+1 = 1729. Therefore 1729 continues the exact same quartic-plus-cubic pattern and is the correct next term.

Option A:
Option A, 1713, is smaller than the formula’s output and does not equal n⁴+2n³+1 for n = 6. It would imply subtracting 16 from the correct value without any justification in the earlier terms. Hence option A violates the established algebraic rule and is incorrect.

Option B:
Option B, 1729, matches precisely the result from the expression n⁴+2n³+1 when n = 6. It maintains the same combination of powers of n that generated all previous numbers in the series. Because no change to the functional relationship is needed, this option correctly extends the sequence.

Option C:
Option C, 1735, slightly overshoots the correct value and again fails to satisfy the expression for n = 6. Choosing 1735 would mean adding an arbitrary 6 only at the last step, which is inconsistent with the pattern. Therefore option C cannot be the correct answer.

Option D:
Option D, 1745, is even further from the predicted value and has no basis in the rule aₙ = n⁴ + 2n³ + 1. Using 1745 would destroy the exact mapping between the index and the term values. Thus option D is not a valid continuation of the given series.

The series is described by the formula aₙ = 2n⁴ + 3n³ + n² with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2+3+1 = 6, 32+24+4 = 60, 162+81+9 = 252, 512+192+16 = 720 and 1250+375+25 = 1650. For n = 6 this expression yields 2·6⁴+3·6³+6² = 2592+648+36 = 3276. Thus 3276 is the unique next term that maintains this quartic-plus-cubic-plus-quadratic structure.

Option A:
Option A, 3210, is smaller than the formula’s result and does not satisfy aₙ = 2n⁴+3n³+n² for n = 6. It would require subtracting 66 from the correctly computed value at the last term only. Because the earlier terms follow the rule exactly, such a modification is unjustified and makes option A incorrect.

Option B:
Option B, 3276, matches precisely the value computed from the expression for n = 6. It preserves all polynomial components and coefficients that describe the growth of the series from the first to the fifth term. Since the same rule generates the sixth term without alteration, 3276 is the correct continuation.

Option C:
Option C, 3330, overshoots the formula’s output and cannot be obtained by substituting n = 6 into the rule. Choosing 3330 would mean arbitrarily inflating the sixth term, which breaks the tight correspondence between index and value. Therefore option C is incorrect.

Option D:
Option D, 3396, deviates even more from the correct value and has no basis in the expression 2n⁴+3n³+n² for the given index. Using 3396 would destroy the internal consistency of the polynomial pattern, so option D 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.