In the number series 2, 11, 57, 288, 1444, the next term is _______. – UGC NET Question

The sequence is defined recursively by a₁ = 2 and aₙ₊₁ = 5aₙ + n for n ≥ 1. Using this rule we get a₂ = 5·2+1 = 11, a₃ = 5·11+2 = 57, a₄ = 5·57+3 = 288 and a₅ = 5·288+4 = 1444, which reproduces the given series. For n = 5 the next term is a₆ = 5·1444+5 = 7220+5 = 7225. Thus 7225 is the only value that fits the same recurrence structure for the sixth term.

Option A:
Option A, 7085, is significantly smaller than the recurrence output and does not equal 5·1444+5. It would require changing the multiplier or subtracting a large amount at the last step, neither of which is indicated by the previous terms. Hence option A is not concordant with the established pattern.

Option B:
Option B, 7155, is closer but still falls short of the correct value 7225, and it is not produced by the rule aₙ₊₁ = 5aₙ+n with n = 5. Choosing 7155 would introduce an unexplained discrepancy between the rule and the sequence. Therefore option B is incorrect.

Option C:
Option C, 7185, again does not satisfy 5·1444+5 and reflects an arbitrary reduction of 40 compared to the formula result. Since the recurrence has been obeyed exactly by all earlier terms, such a modification is unjustified. Option C is therefore incorrect.

Option D:
Option D, 7225, equals the exact value given by the recurrence for the sixth term. It continues the pattern of multiplying the previous term by 5 and adding the index, which explains all of the earlier numbers. Because it fits perfectly with the recursive rule, 7225 is the correct next term in the series.