In the number series 2, 18, 84, 260, 630, the next term is _______. – UGC NET Question

This series is generated by the formula (a_n = n^4 + n) for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 1+1 = 2, 16+2 = 18, 81+3 = 84, 256+4 = 260 and 625+5 = 630, which match the given terms. For n = 6 the same rule yields (6^4 + 6 = 1296 + 6 = 1302). Hence 1302 is the unique next term that respects this quartic-plus-linear pattern.

Option A:
Option A, 1286, is 16 less than the formula’s output and does not equal (6^4 + 6). It would require altering the rule arbitrarily for the last term, which is not supported by the earlier data. Therefore 1286 is not correct.

Option B:
Option B, 1294, is closer but still not equal to 1302 and fails to satisfy the expression (n^4 + n) for n = 6. Choosing 1294 would weaken the exact algebraic structure of the series. Thus 1294 is not a valid continuation.

Option C:
Option C, 1310, overshoots the correct value and again cannot be written as (6^4 + 6). Adopting 1310 would break the close correspondence between the formula and the actual terms. Therefore 1310 is not acceptable.

Option D:
Option D, 1302, equals (6^4 + 6) exactly and continues the same quartic pattern without modification. It maintains the functional relationship that has generated every earlier term, so 1302 is the correct next term.