In the number series 3, 8, 21, 64, 265, the next term is _______. – UGC NET Question

The pattern can be described by aₙ = 2n! + n² with n starting from 1. For n = 1, 2, 3, 4 and 5, this yields 2·1! + 1² = 3, 2·2! + 2² = 8, 2·3! + 3² = 21, 2·4! + 4² = 64 and 2·5! + 5² = 265, matching the given terms. For n = 6, we obtain a₆ = 2·6! + 6² = 2·720 + 36 = 1476. Thus 1476 is the only value that fits this factorial-based rule and continues the series.

Option A:
Option A, 1452, is 24 less than the correct value given by 2n! + n² at n = 6. To pick 1452 we would need to subtract 24 from the computed result only at this position, which would disrupt the pattern of combining factorial and square terms. Therefore option A is incorrect.

Option B:
Option B, 1476, exactly matches the number produced by 2·6! + 6². It preserves the factorial component and the quadratic addition in the same way as for earlier values of n. Because it extends the rule without any modification, this option correctly continues the series.

Option C:
Option C, 1464, is 12 smaller than the correct value and cannot be expressed as 2·720 + 36. It is only an approximate candidate and does not reflect the precise functional relationship. As such, option C is not a valid next term.

Option D:
Option D, 1488, is 12 greater than the required value and again fails to satisfy the formula at n = 6. Choosing 1488 would overstate the term in comparison to what the rule dictates, breaking the pattern. Hence option D is not acceptable.