In the number series 4, 31, 158, 505, 1240, the next term is _______. – UGC NET Question

The series satisfies the rule (a_n = 2n^4 - 3n + 5) with n starting from 1. For n = 1, 2, 3, 4 and 5 we get 2−3+5 = 4, 32−6+5 = 31, 162−9+5 = 158, 512−12+5 = 505 and 1250−15+5 = 1240. For n = 6 we compute (2×6^4 - 18 + 5 = 2592 - 13 = 2579). Hence 2579 is the correct next term.

Option A:
Option A, 2543, is 36 less than the value predicted by the rule and does not equal (2×6^4 - 3×6 + 5). It introduces an arbitrary shortfall not reflected earlier, so 2543 is not valid.

Option B:
Option B, 2561, is closer but still fails to match 2579 and cannot be derived from the formula for n = 6. Choosing 2561 would alter the established pattern. Thus 2561 is not correct.

Option C:
Option C, 2597, overshoots the computed value and again does not satisfy the expression for the next index. Using 2597 would inflate the term without justification and break the quartic structure. Hence 2597 is not appropriate.

Option D:
Option D, 2579, equals the exact output of (a_n = 2n^4 - 3n + 5) when n = 6. It maintains the same relationship between index and term for all entries in the series and is therefore the correct continuation.