In the number series 7, 18, 47, 124, 333, the next term is _______. – UGC NET Question

The pattern can be represented by a₁ = 7 and aₙ₊₁ = 2aₙ + 3ⁿ + n for n ≥ 1. Applying this rule, we have a₂ = 2·7 + 3¹ + 1 = 18, a₃ = 2·18 + 3² + 2 = 47, a₄ = 2·47 + 3³ + 3 = 124 and a₅ = 2·124 + 3⁴ + 4 = 333, which coincides with the given terms. For n = 5 the next term is a₆ = 2·333 + 3⁵ + 5 = 666 + 243 + 5 = 914. Therefore 914 is the unique value that satisfies this recurrence and extends the sequence.

Option A:
Option A, 890, is 24 less than the recurrence value at n = 5 and cannot be obtained from 2a₅ + 3⁵ + 5. Choosing 890 would require subtracting 24 from the correctly computed term only at this point. This change breaks the exact rule defining the series, so option A is not correct.

Option B:
Option B, 914, exactly matches the number produced by the recurrence when a₅ = 333 and n = 5. It preserves the doubling of the previous term together with the exponential component and linear adjustment. Because this description explains each step in the sequence, 914 is the correct next term.

Option C:
Option C, 902, is 12 less than the required value and does not equal 2·333 + 243 + 5. It approximates the correct term but fails to follow from the specified recurrence relation. Hence option C cannot be accepted as the continuation.

Option D:
Option D, 926, is 12 greater than the correct value and again fails to satisfy the rule at n = 5. To get 926 we would need to inflate the additive part, altering the recurrence at the final step. Thus option D is not a valid extension of the number series.