In the number series 4, 19, 61, 153, 371, the next term is _______. – UGC NET Question

This series can be expressed as aₙ = n! + 2n³ + 1 with n starting from 1. Computing the values, we get for n = 1, 2, 3, 4 and 5: 1! + 2·1³ + 1 = 4, 2! + 2·2³ + 1 = 19, 3! + 2·3³ + 1 = 61, 4! + 2·4³ + 1 = 153 and 5! + 2·5³ + 1 = 371, which matches the sequence. For n = 6, a₆ = 6! + 2·6³ + 1 = 720 + 432 + 1 = 1153. Therefore 1153 is the unique number that continues this factorial-cubic pattern.

Option A:
Option A, 1153, is exactly the value obtained from 6! + 2·6³ + 1. It preserves the factorial growth together with the cubic and constant components that determine the earlier terms. Since no change to the rule is needed at the sixth term, this option correctly extends the series.

Option B:
Option B, 1129, is 24 less than the correct value and cannot be produced by n! + 2n³ + 1 at n = 6. Accepting 1129 would require reducing the computed term without any justification from the pattern. Hence option B is not a valid continuation.

Option C:
Option C, 1141, is 12 smaller than 1153 and again does not equal 720 + 432 + 1. It approximates the correct answer but does not arise from the formula. Thus option C fails to accurately follow the sequence rule.

Option D:
Option D, 1165, is 12 greater than the computed value and would demand increasing the expression’s output only at the sixth position. This breaks the exact correspondence between the formula and the data, so option D is incorrect.