In the number series 11, 44, 139, 356, 779, the next term is _______. – UGC NET Question

This sequence follows the rule aₙ = n⁴ + 6n² + 4 with n starting from 1. Evaluating the expression gives 1+6+4 = 11, 16+24+4 = 44, 81+54+4 = 139, 256+96+4 = 356 and 625+150+4 = 779, which confirms the pattern. For n = 6 we obtain 6⁴+6·6²+4 = 1296+216+4 = 1516. Hence 1516 is the only value that continues this quartic-plus-quadratic-plus-constant structure.

Option A:
Option A, 1516, equals the value given by n⁴+6n²+4 for n = 6. It preserves the same coefficients for both the quartic and quadratic terms and keeps the constant term unchanged. Because the formula explains every term including the next, this option correctly extends the series.

Option B:
Option B, 1484, is 32 less than the correct polynomial value and does not satisfy the expression for n = 6. Choosing 1484 would require an arbitrary reduction at the last term that contradicts the earlier behaviour of the sequence. Therefore option B is incorrect.

Option C:
Option C, 1532, overshoots the formula’s result by 16 and again does not arise from substituting n = 6 into the rule. It implies a modification of the constant or coefficients, which the earlier terms do not support. Hence option C is incorrect.

Option D:
Option D, 1556, deviates even more from 1516 and has no direct basis in the function aₙ = n⁴+6n²+4. Using 1556 would break the precise match between index and term that defines the series, so option D is incorrect.