In the number series 4, 50, 246, 772, 1880, the next term is _______. – UGC NET Question

The terms satisfy the rule (a_n = 3n^4 + n) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 3+1 = 4, 48+2 = 50, 243+3 = 246, 768+4 = 772 and 1875+5 = 1880. For n = 6 the expression gives (3×6^4 + 6 = 3×1296 + 6 = 3888 + 6 = 3894). Hence 3894 is the term that keeps this quartic-plus-linear pattern intact.

Option A:
Option A, 3846, is 48 less than the computed value and does not equal (3×6^4 + 6). It would require unjustified modification of the rule at the final term. Therefore 3846 is not a correct continuation.

Option B:
Option B, 3862, is closer but still fails to match 3894 and does not arise from the formula for n = 6. Choosing 3862 would weaken the exact algebraic structure that defines the sequence. Hence 3862 is not valid.

Option C:
Option C, 3894, comes directly from evaluating (a_n = 3n^4 + n) at n = 6. It preserves both the quartic and linear components exactly as before, ensuring internal consistency. For this reason, 3894 is the correct next term.

Option D:
Option D, 3950, overshoots the correct value and again cannot be written as (3n^4 + n) for the next index. Using 3950 would distort the pattern, so it is not the right answer.