In the number series 7, 42, 137, 322, 627, the next term is _______. – UGC NET Question

The sequence follows the rule (a_n = 5n^3 + 2) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 7, 42, 137, 322 and 627, which agree with the given terms. For n = 6 the formula gives (5×6^3 + 2 = 5×216 + 2 = 1080 + 2 = 1082). Thus 1082 is the only value that continues this cubic pattern exactly.

Option A:
Option A, 1082, is exactly the value produced by (a_n = 5n^3 + 2) when n = 6. It keeps the same coefficient and constant term that explain all previous terms. Because it arises naturally from the rule, 1082 is the correct next term in the series.

Option B:
Option B, 1066, is 16 less than the computed value and does not satisfy the cubic expression. It would suggest a sudden decrease not supported by the pattern. Hence 1066 is not a valid continuation.

Option C:
Option C, 1094, overshoots the formula’s output and cannot be obtained from (5n^3 + 2) for the next index. Selecting 1094 would arbitrarily increase the term and disrupt the algebraic structure. Therefore 1094 is not correct.

Option D:
Option D, 1106, is even further away from the expected value and similarly fails to equal (5×6^3 + 2). Using 1106 would break the strong consistency of the sequence, so it is not acceptable.