In the number series 7, 29, 124, 523, 2156, the next term is _______. – UGC NET Question

The series can be described recursively by a₁ = 7 and aₙ₊₁ = 4aₙ + n³ for n ≥ 1. Using this rule we obtain a₂ = 4·7+1³ = 29, a₃ = 4·29+2³ = 124, a₄ = 4·124+3³ = 523 and a₅ = 4·523+4³ = 2156, which reproduces the given terms. For n = 5 the next term is a₆ = 4·2156+5³ = 8624+125 = 8749. Therefore 8749 is the unique value that continues the same recurrence for the sixth term.

Option A:
Option A, 8605, is less than the recurrence output and does not equal 4·2156+5³. It would correspond to adding -19 to 4·2156 instead of the required 125, which contradicts the index-dependent cubic correction. Hence option A is incorrect.

Option B:
Option B, 8681, is closer but still fails to reach the correct value 8749 produced by the recurrence. Selecting 8681 would suggest a smaller adjustment by 57 instead of adding 125, again violating the rule aₙ₊₁ = 4aₙ+n³. Thus option B is incorrect.

Option C:
Option C, 8717, differs by 32 from the computed value and cannot be obtained from 4·2156+5³. It implies a partial use of the cubic term, which is not how the earlier terms were formed. Therefore option C is incorrect.

Option D:
Option D, 8749, matches exactly the result of applying the recurrence to a₅ with n = 5. It maintains the structure of quadrupling the previous term and then adding the cube of the index. Because this rule has generated all previous terms, 8749 is the correct next term in the series.