In the number series 2, 24, 108, 320, 750, the next term is _______. – UGC NET Question

This series is governed by the formula aₙ = n⁴+n³ for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1+1 = 2, 16+8 = 24, 81+27 = 108, 256+64 = 320 and 625+125 = 750. For n = 6 the expression gives 1296+216 = 1512. Therefore 1512 is the only value that maintains the quartic-plus-cubic structure of the sequence.

Option A:
Option A, 1432, is 80 less than the correct value and does not equal n⁴+n³ for n = 6. It breaks the clear algebraic rule that generates the previous terms. Thus 1432 cannot be the correct continuation.

Option B:
Option B, 1512, is exactly the sum of 6⁴ and 6³ and follows directly from the rule aₙ = n⁴+n³. It preserves both the degree and the coefficients of the terms in the expression. For these reasons, 1512 is the correct next term.

Option C:
Option C, 1462, also fails to satisfy the formula for n = 6 and would require altering the mathematical relationship. This inconsistency makes 1462 an invalid choice.

Option D:
Option D, 1572, overshoots the predicted value and again cannot be obtained from n⁴+n³ for the next index. Choosing 1572 would abandon the tight functional fit, so it is not correct.