In the number series 3, 26, 111, 324, 755, the next term is _______. – UGC NET Question

This sequence is given by the formula (a_n = n^4 + n^3 + n) for n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 1+1+1 = 3, 16+8+2 = 26, 81+27+3 = 111, 256+64+4 = 324 and 625+125+5 = 755. For n = 6 we obtain (6^4 + 6^3 + 6 = 1296 + 216 + 6 = 1518). Therefore 1518 is the correct next term.

Option A:
Option A, 1494, is smaller than the formula’s output and does not equal (6^4 + 6^3 + 6). It would impose an unjustified reduction at the last step, so 1494 is not a valid continuation.

Option B:
Option B, 1506, is still below 1518 and fails to satisfy the expression for n = 6. Choosing 1506 would break the quartic-plus-cubic structure of the sequence. Thus 1506 is not correct.

Option C:
Option C, 1518, matches exactly the value produced by the rule when n = 6. It preserves the same combination of quartic and cubic terms that generated all the earlier numbers and hence is the correct next term.

Option D:
Option D, 1530, overshoots the computed value and cannot be written as (n^4 + n^3 + n) for the next index. Using 1530 would distort the carefully balanced pattern, so it is not appropriate.