In the number series 1, 34, 169, 526, 1273, the next term is _______. – UGC NET Question

The sequence is governed by aₙ = 2n⁴ + n² − 2 for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 2·1⁴+1²−2 = 1, 2·2⁴+2²−2 = 34, 2·3⁴+3²−2 = 169, 2·4⁴+4²−2 = 526 and 2·5⁴+5²−2 = 1273, confirming the rule. For n = 6 we have 2·6⁴+6²−2 = 2·1296+36−2 = 2592+36−2 = 2626. Thus 2626 is the unique value that continues this quartic-based pattern.

Option A:
Option A, 2594, is 32 less than the value mandated by the formula and does not satisfy 2n⁴+n²−2 for n = 6. It would require an unexplained reduction in the final term relative to the established rule. Therefore this option is not compatible with the structure of the sequence.

Option B:
Option B, 2610, is closer but still not equal to 2626 and again fails to arise from substituting n = 6 into the expression. Choosing 2610 would modify the pattern only at the last point, which is mathematically unsound. Hence option B is not correct.

Option C:
Option C, 2626, exactly matches the computed value from aₙ = 2n⁴+n²−2 when n = 6. It preserves the balance between the quartic and quadratic components that generated the earlier terms. Because it follows naturally from the same algebraic rule, 2626 is the correct next term in the series.

Option D:
Option D, 2642, overshoots the formula’s output by 16 and does not respect the rule 2n⁴+n²−2. Selecting 2642 would break the precise connection between term position and term value. Therefore option D cannot be accepted as a valid continuation.