In the number series 3, 34, 168, 536, 1370, the next term is _______. – UGC NET Question

This sequence can be modelled by aₙ = 2n⁴ + n! with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2·1⁴ + 1! = 3, 2·2⁴ + 2! = 34, 2·3⁴ + 3! = 168, 2·4⁴ + 4! = 536 and 2·5⁴ + 5! = 1370, which matches the given terms. For n = 6 we have a₆ = 2·6⁴ + 6! = 2·1296 + 720 = 2592 + 720 = 3312. Hence 3312 is the unique value that maintains this factorial-quartic pattern.

Option A:
Option A, 3288, is 24 less than the value produced by 2n⁴ + n! at n = 6. Choosing 3288 would mean reducing the correct outcome of the formula only at the final term, which is not supported by earlier behaviour. Therefore option A is not consistent with the established rule.

Option B:
Option B, 3300, is 12 smaller than the correct value and cannot be written as 2·1296 + 720. It is numerically close but does not respect the exact functional dependence on n. As a result, option B does not correctly continue the series.

Option C:
Option C, 3324, is 12 greater than the correct term and again cannot be obtained from 2·6⁴ + 6!. Accepting 3324 would force an unjustified increase in the last value relative to the pattern. Thus option C is not a valid continuation.

Option D:
Option D, 3312, matches exactly the number generated by the rule aₙ = 2n⁴ + n! for n = 6. It preserves the relationship that combines a strong quartic component with factorial growth evident in previous terms. Because it extends the same pattern smoothly, this option is the correct answer.