In the number series 4, 18, 76, 312, 1264, the next term is _______. – UGC NET Question

This sequence is defined recursively by a₁ = 4 and aₙ₊₁ = 4aₙ + 2ⁿ for n ≥ 1. Applying the rule gives a₂ = 4·4 + 2¹ = 18, a₃ = 4·18 + 2² = 76, a₄ = 4·76 + 2³ = 312 and a₅ = 4·312 + 2⁴ = 1264, exactly matching the given terms. For n = 5 the next term is a₆ = 4·1264 + 2⁵ = 5088. Therefore 5088 is the unique number that satisfies the recurrence and continues the series.

Option A:
Option A, 5056, is 32 less than the value given by aₙ₊₁ = 4aₙ + 2ⁿ when n = 5. Accepting 5056 would mean reducing the correct result by the size of the power term 2⁵, breaking the pattern of adding that full power at each stage. Hence option A does not follow the established rule and is incorrect.

Option B:
Option B, 5088, matches exactly the value obtained from 4a₅ + 2⁵ with a₅ = 1264. It maintains the structure of multiplying the previous term by four and then adding the appropriate power of 2. Because this mechanism generates all earlier terms and leads to 5088 next, option B correctly continues the sequence.

Option C:
Option C, 5104, is 16 greater than the recurrence value and cannot be produced by 4a₅ + 2⁵. It would require adding more than the specified power of 2 at the final step, contradicting the rule. Thus option C does not preserve the exact relationship and is not valid.

Option D:
Option D, 5120, deviates even more and again fails to equal 4·1264 + 32. Using 5120 would ignore the precise recurrence that describes the sequence. Therefore option D is not a correct continuation.