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.

In this sequence each term is given by aₙ = n! + n² for n starting from 1. Checking this rule, we get 1!+1² = 2, 2!+2² = 6, 3!+3² = 15, 4!+4² = 40 and 5!+5² = 145, which matches the given terms exactly. For n = 6 the expression yields 6!+6² = 720+36 = 756. Consequently, 756 is the only term that continues the factorial-plus-square pattern.

Option A:
Option A, 720, corresponds to 6! alone and omits the n² part of the expression. It does not follow the rule aₙ = n!+n² that has generated all previous terms. Therefore option A ignores part of the pattern and cannot be accepted as correct.

Option B:
Option B, 744, adds only 24 to 720 instead of the required 36, so it does not equal 6!+6². Selecting 744 would require changing the square component from n² to a smaller value, which is inconsistent with the earlier terms. Hence option B is not a valid continuation.

Option C:
Option C, 756, matches exactly the value of 6!+6² and maintains both components of the rule: the factorial growth and the quadratic correction. It fits the behaviour observed for n = 1 to 5, where each term is one factorial plus a square. Because it preserves this combined structure, 756 is the correct next term in the series.

Option D:
Option D, 780, adds 60 to 720 and cannot be written as n!+n² for n = 6. It would imply using a different function of n for the second part of the expression. This breaks the pattern evident from the given terms, so option D is incorrect.