In the number series 6, 42, 164, 450, 1002, the next term is _______. – UGC NET Question

The terms fit the polynomial rule aₙ = n⁴ + 3n³ + 2 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 into this expression yields 6, 42, 164, 450 and 1002, exactly the numbers in the series. Evaluating the same formula for n = 6 produces 1946. Hence 1946 is the only value that extends this quartic-plus-cubic pattern correctly.

Option A:
Option A, 1922, is 24 less than the value obtained from aₙ = n⁴ + 3n³ + 2 when n = 6. Accepting 1922 would require reducing the polynomial result only at the sixth term, which contradicts the behaviour of earlier entries. Therefore option A is not consistent with the established pattern.

Option B:
Option B, 1934, is still lower than the required value and does not equal the expression’s output at n = 6. It lies between the earlier predictions but fails to respect the exact algebraic rule. As a result, option B cannot be the correct continuation of the sequence.

Option C:
Option C, 1946, matches precisely the value produced by the formula when n = 6. It preserves the same contributions from the fourth-power and cubic terms together with the constant part. Because this rule explains all given terms and leads naturally to 1946, option C is the correct answer.

Option D:
Option D, 1962, is 16 greater than the polynomial value and cannot be generated by n⁴ + 3n³ + 2 at n = 6. Introducing such an extra increment would break the clean functional relationship. Hence option D is not acceptable.