UGC NET Questions (Paper – 1)

The sequence is generated by aₙ = n⁴ + n³ + 3n² + 2 for n starting from 1. For n = 1, 2, 3, 4 and 5 we calculate 1+1+3+2 = 7, 16+8+12+2 = 38, 81+27+27+2 = 137, 256+64+48+2 = 370 and 625+125+75+2 = 827, matching the given terms exactly. For n = 6 this formula yields 6⁴+6³+3·6²+2 = 1296+216+108+2 = 1622. Thus 1622 is the only value that continues this specific quartic-plus-cubic-plus-quadratic pattern.

Option A:
Option A, 1622, matches perfectly the value obtained from n⁴+n³+3n²+2 when n = 6. It preserves all components of the expression and continues the same growth trend seen in the previous terms. Because no modification of the coefficients or constant is required, 1622 is the correct continuation of the series.

Option B:
Option B, 1606, is 16 less than the formula’s output and does not satisfy the rule for n = 6. Choosing 1606 would require reducing the term without any structural change in the function, which contradicts the exact fit observed earlier. Hence option B is incorrect.

Option C:
Option C, 1640, overshoots the correct value by 18 and again cannot be written as n⁴+n³+3n²+2 for n = 6. It represents an arbitrary increase that is not indicated by the previous behaviour of the series. Therefore option C is incorrect.

Option D:
Option D, 1664, deviates even more from the computed value and has no basis in the established expression. Using 1664 would break the precise mapping between index and term that defines this number series, so option D is incorrect.