UGC NET Questions (Paper – 1)

The pattern can be expressed recursively with a₁ = 6 and aₙ₊₁ = 3aₙ + 2ⁿ + n for n ≥ 1. Using this rule we get a₂ = 3·6 + 2¹ + 1 = 21, a₃ = 3·21 + 2² + 2 = 69, a₄ = 3·69 + 2³ + 3 = 218 and a₅ = 3·218 + 2⁴ + 4 = 674, which matches the given series. For n = 5 the next term is a₆ = 3·674 + 2⁵ + 5 = 2059. Thus 2059 is the only value consistent with this recurrence.

Option A:
Option A, 2059, is exactly the value obtained from aₙ₊₁ = 3aₙ + 2ⁿ + n when applied to a₅ with n = 5. It keeps the structure of tripling the previous term, adding a power of 2 and then adding the index itself. Because this mechanism reproduces all previous terms and yields 2059 next, this option is correct.

Option B:
Option B, 2035, is 24 less than the recurrence result and cannot be written as 3·674 + 2⁵ + 5. Accepting 2035 would require subtracting 24 from the properly computed term, with no justification from the earlier steps. Therefore this option does not respect the established pattern and is incorrect.

Option C:
Option C, 2073, is 14 greater than the required value and also fails to match the recurrence. To reach 2073 the additive part after 3a₅ would have to be larger than 2⁵ + 5, changing the rule at the last step. Consequently, option C is not a valid continuation.

Option D:
Option D, 2087, deviates even more from 2059 and likewise cannot be produced by the rule. Using 2087 would break the exact connection between consecutive terms that defines the sequence. Hence option D cannot be the correct answer.

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.

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.