In the number series 1, 31, 161, 511, 1249, the next term is _______. – UGC NET Question

The pattern here arises from the formula aₙ = 2n⁴−1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 2·1⁴−1 = 1, 2·2⁴−1 = 31, 2·3⁴−1 = 161, 2·4⁴−1 = 511 and 2·5⁴−1 = 1249. For n = 6 we compute 2·6⁴−1 = 2·1296−1 = 2592−1 = 2591. Therefore 2591 is the only term that correctly extends the quartic-based pattern.

Option A:
Option A, 2511, is far below the calculated value and does not equal 2n⁴−1 for any n that continues the sequence. It ignores the rapid growth implied by the quartic expression. Thus 2511 is not a valid next term.

Option B:
Option B, 2541, is still different from 2591 and cannot be obtained from 2·6⁴−1. Choosing it would break the exact algebraic rule that explains all the previous terms. Hence 2541 cannot be the correct answer.

Option C:
Option C, 2591, matches the result of evaluating 2n⁴−1 when n = 6. It preserves the quartic dependence on n and continues the rapid increase seen in the sequence. For these reasons, 2591 is the correct continuation.

Option D:
Option D, 2651, overshoots the correct value and is not equal to 2n⁴−1 for any appropriate n in order. Adopting 2651 would distort the pattern and introduce an unjustified jump. Therefore it is not a valid option.