In the number series 7, 22, 70, 219, 673, the next term is _______. – UGC NET Question

The sequence can be defined recursively by a₁ = 7 and aₙ₊₁ = 3aₙ + n² for n ≥ 1. Using this rule, a₂ = 3·7 + 1² = 22, a₃ = 3·22 + 2² = 70, a₄ = 3·70 + 3² = 219 and a₅ = 3·219 + 4² = 673, matching the given terms. For n = 5 the next term is a₆ = 3·673 + 5² = 2044. Thus 2044 is the only value that satisfies the same recurrence and correctly continues the series.

Option A:
Option A, 2044, is exactly the result produced by aₙ₊₁ = 3aₙ + n² when it is applied to a₅ with n = 5. It preserves the pattern of tripling the previous term and adding the square of the index. Because this rule explains each step of the sequence and yields 2044 next, this option is correct.

Option B:
Option B, 2032, is 12 less than the recurrence output for n = 5. To accept 2032 we would have to arbitrarily subtract 12 from the correctly computed value while keeping the rule unchanged before. Such a disturbance is not justified by the earlier progression, so option B is incorrect.

Option C:
Option C, 2056, is 12 more than the required value from the recurrence. Selecting 2056 would similarly require increasing the formula result only at this final step. That would break the precise relationship described by aₙ₊₁ = 3aₙ + n², making option C invalid.

Option D:
Option D, 2072, deviates even further from the recurrence value and cannot be written as 3·673 + 5². Using 2072 would disregard the established rule and destroy the exact fit observed in earlier terms. Therefore this option does not correctly continue the series.