In the number series 5, 23, 102, 437, 1814, the next term is _______. – UGC NET Question

The pattern can be written as a₁ = 5 and aₙ₊₁ = 4aₙ + n³ + 2 for n ≥ 1. Applying this rule gives a₂ = 4·5 + 1³ + 2 = 23, a₃ = 4·23 + 2³ + 2 = 102, a₄ = 4·102 + 3³ + 2 = 437 and a₅ = 4·437 + 4³ + 2 = 1814, all of which match the series. For n = 5 the next term is a₆ = 4·1814 + 5³ + 2 = 7256 + 125 + 2 = 7383. Thus 7383 is the only value that satisfies this recurrence and correctly continues the sequence.

Option A:
Option A, 7359, is 24 less than the recurrence output for n = 5 and cannot be written as 4a₅ + 5³ + 2. Accepting 7359 would require subtracting 24 from the correctly computed term only at the final step. This violates the consistency of the recurrence, so option A is incorrect.

Option B:
Option B, 7371, is 12 smaller than the correct value 7383 and likewise does not equal 4·1814 + 125 + 2. It provides an approximate but not exact continuation and therefore does not follow the specified rule. Consequently, option B cannot be chosen as the next term.

Option C:
Option C, 7395, is 12 greater than the correct value and again fails to match the recurrence formula. To obtain 7395 we would have to add extra units to the computed result at n = 5, altering the defined update pattern. Hence option C is not a valid extension of the sequence.

Option D:
Option D, 7383, exactly equals the value produced by the rule aₙ₊₁ = 4aₙ + n³ + 2 when n = 5. It preserves the combination of quadrupling the previous term with an added cubic and constant adjustment. Because this structure explains every step of the series, 7383 is the correct answer.