In the number series 1, 10, 45, 136, 325, the next term is _______. – UGC NET Question

The series can be expressed by the formula aₙ = (n⁴+n²)/2 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get (1+1)/2 = 1, (16+4)/2 = 10, (81+9)/2 = 45, (256+16)/2 = 136 and (625+25)/2 = 325, matching the terms given. For n = 6 the expression gives (6⁴+6²)/2 = (1296+36)/2 = 1332/2 = 666. Thus 666 is the unique next term that fits this quartic-based sequence.

Option A:
Option A, 646, is 20 less than the computed value and does not equal (6⁴+6²)/2. It has no basis in the formula that successfully models all the earlier terms. Therefore 646 cannot serve as the correct continuation.

Option B:
Option B, 654, is still short of 666 and similarly fails to arise from the expression involving n⁴ and n². Choosing 654 would force a change in the pattern for the last term only, which is mathematically unsound. Hence it is not the right answer.

Option C:
Option C, 681, is larger than the predicted value and again does not match (6⁴+6²)/2. It represents an arbitrary modification rather than a logical extension of the rule. Thus 681 is not a valid next term.

Option D:
Option D, 666, exactly matches the output of the formula when n = 6. It preserves the structural relationship between the terms and their positions and keeps the series fully consistent. For this reason, 666 is the correct option.