In the number series 4, 33, 136, 385, 876, the next term is _______. – UGC NET Question

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.