UGC NET Questions (Paper – 1)

Here the pattern is aₙ = n⁴ + 2n³ + 1 with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1+2+1 = 4, 16+16+1 = 33, 81+54+1 = 136, 256+128+1 = 385 and 625+250+1 = 876, which matches the sequence exactly. For n = 6 we compute 6⁴+2·6³+1 = 1296+432+1 = 1729. Therefore 1729 continues the exact same quartic-plus-cubic pattern and is the correct next term.

Option A:
Option A, 1713, is smaller than the formula’s output and does not equal n⁴+2n³+1 for n = 6. It would imply subtracting 16 from the correct value without any justification in the earlier terms. Hence option A violates the established algebraic rule and is incorrect.

Option B:
Option B, 1729, matches precisely the result from the expression n⁴+2n³+1 when n = 6. It maintains the same combination of powers of n that generated all previous numbers in the series. Because no change to the functional relationship is needed, this option correctly extends the sequence.

Option C:
Option C, 1735, slightly overshoots the correct value and again fails to satisfy the expression for n = 6. Choosing 1735 would mean adding an arbitrary 6 only at the last step, which is inconsistent with the pattern. Therefore option C cannot be the correct answer.

Option D:
Option D, 1745, is even further from the predicted value and has no basis in the rule aₙ = n⁴ + 2n³ + 1. Using 1745 would destroy the exact mapping between the index and the term values. Thus option D is not a valid continuation of the given series.

This series is generated by the quintic rule aₙ = n⁵ − n + 1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 1−1+1 = 1, 32−2+1 = 31, 243−3+1 = 241, 1024−4+1 = 1021 and 3125−5+1 = 3121, exactly matching the given numbers. For n = 6 the same expression gives 6⁵−6+1 = 7776−6+1 = 7771. Therefore 7771 is the unique next term that maintains this fifth-degree pattern.

Option A:
Option A, 7729, is significantly below the computed value and does not satisfy aₙ = n⁵−n+1 for n = 6. It would require subtracting 42 from the formula’s result only at the last term. Such a deviation has no basis in the earlier behaviour of the sequence, so option A is incorrect.

Option B:
Option B, 7759, is closer but still does not equal 7771 and therefore fails to arise from applying n⁵−n+1 to n = 6. Selecting 7759 would alter the functional rule that perfectly explains all previous terms. Hence option B is not a valid continuation of the pattern.

Option C:
Option C, 7771, coincides exactly with 6⁵−6+1 and follows directly from the quintic expression. It keeps the same high-order dependence on n that produces the rapid growth observed in the sequence. Because it continues the rule without modification, 7771 is the correct next term.

Option D:
Option D, 7789, overshoots the formula’s output by 18 and cannot be expressed as n⁵−n+1 for the sixth position. Using 7789 would break the precise relationship between index and term, making option D inconsistent with the established series.

The series can be modelled by the recurrence a₁ = 4 and aₙ₊₁ = 2aₙ + 3n² − 1 for n ≥ 1. Applying this rule gives a₂ = 2·4+3·1²−1 = 10, a₃ = 2·10+3·2²−1 = 31, a₄ = 2·31+3·3²−1 = 88 and a₅ = 2·88+3·4²−1 = 223, which matches the given sequence. For n = 5 the next term is a₆ = 2·223+3·5²−1 = 446+75−1 = 520. Therefore 520 is the unique next term that obeys this recurrence.

Option A:
Option A, 520, is exactly the value produced by the rule aₙ₊₁ = 2aₙ+3n²−1 when applied to a₅ = 223 with n = 5. It preserves the same structure of doubling the previous term and adding a square-based correction. Since all previous terms are generated in exactly this way, 520 is the correct continuation.

Option B:
Option B, 512, is slightly smaller than the recurrence output and does not satisfy 2·223+3·5²−1. It would require subtracting 8 from the correct value only at the final step. This arbitrary change is not reflected earlier in the sequence, so option B is not consistent with the pattern.

Option C:
Option C, 528, is larger than the correctly computed value and cannot be derived from the recurrence without altering its parameters. Choosing 528 would artificially inflate the sixth term and destroy the exact match between the rule and the data. Hence option C is incorrect.

Option D:
Option D, 536, deviates even more from 520 and again fails to arise from the relation aₙ₊₁ = 2aₙ+3n²−1. Using 536 would break the carefully defined index-dependent correction term, so this option is incorrect.

This series follows the rule aₙ = 2n⁴ + 5n² + 3 with n starting from 1. Substituting n = 1, 2, 3, 4 and 5 yields 2+5+3 = 10, 32+20+3 = 55, 162+45+3 = 210, 512+80+3 = 595 and 1250+125+3 = 1378, which matches the given numbers. For n = 6 we compute 2·6⁴+5·6²+3 = 2592+180+3 = 2775. Therefore 2775 is the unique next term consistent with this quartic-plus-quadratic-plus-constant structure.

Option A:
Option A, 2715, is 60 less than the formula output and does not satisfy aₙ = 2n⁴+5n²+3 for n = 6. It introduces an arbitrary downward adjustment that is not indicated by the previous terms. Hence option A is incorrect.

Option B:
Option B, 2775, equals exactly the value obtained by applying the expression to n = 6. It preserves the same coefficients and constant term that were used to generate all earlier entries in the series. Because it fits seamlessly into the algebraic rule, 2775 is the correct next term.

Option C:
Option C, 2835, overshoots the correct value by 60 and again does not arise from substituting n = 6 into the rule. Choosing 2835 would inflate the sixth term without any support from the structure of the series. Therefore option C is incorrect.

Option D:
Option D, 2895, deviates even more from the computed value and has no foundation in the expression 2n⁴+5n²+3. Using 2895 would destroy the precise link between index and term, so option D is incorrect.

The series is described by the formula aₙ = 2n⁴ + 3n³ + n² with n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2+3+1 = 6, 32+24+4 = 60, 162+81+9 = 252, 512+192+16 = 720 and 1250+375+25 = 1650. For n = 6 this expression yields 2·6⁴+3·6³+6² = 2592+648+36 = 3276. Thus 3276 is the unique next term that maintains this quartic-plus-cubic-plus-quadratic structure.

Option A:
Option A, 3210, is smaller than the formula’s result and does not satisfy aₙ = 2n⁴+3n³+n² for n = 6. It would require subtracting 66 from the correctly computed value at the last term only. Because the earlier terms follow the rule exactly, such a modification is unjustified and makes option A incorrect.

Option B:
Option B, 3276, matches precisely the value computed from the expression for n = 6. It preserves all polynomial components and coefficients that describe the growth of the series from the first to the fifth term. Since the same rule generates the sixth term without alteration, 3276 is the correct continuation.

Option C:
Option C, 3330, overshoots the formula’s output and cannot be obtained by substituting n = 6 into the rule. Choosing 3330 would mean arbitrarily inflating the sixth term, which breaks the tight correspondence between index and value. Therefore option C is incorrect.

Option D:
Option D, 3396, deviates even more from the correct value and has no basis in the expression 2n⁴+3n³+n² for the given index. Using 3396 would destroy the internal consistency of the polynomial pattern, so option D is incorrect.

The series is generated by the rule aₙ = n⁴ + 3n, where n is the position of the term starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 1⁴+3·1 = 4, 2⁴+3·2 = 22, 3⁴+3·3 = 90, 4⁴+3·4 = 268 and 5⁴+3·5 = 640, which matches the given terms exactly. For n = 6, the same formula yields 6⁴+3·6 = 1296+18 = 1314. Thus 1314 is the only value that continues the established functional pattern without any adjustment.

Option A:
Option A, 1304, is close to the expected value but does not equal 6⁴+3·6. Choosing 1304 would mean reducing the correct formula output by 10, which is not supported by the behaviour of earlier terms. Therefore this option breaks the consistent polynomial rule and cannot be correct.

Option B:
Option B, 1308, also fails to match the value obtained from n⁴+3n for n = 6. It would require introducing an arbitrary correction of −6 to the formula result only at the last step. Because the first five terms follow the rule exactly, such a change is logically inconsistent with the observed pattern.

Option C:
Option C, 1314, matches exactly the value of 6⁴+3·6 and therefore respects the rule used for all previous terms. It maintains the same dependence on the index n without any modification or exception. Since every term including the next one can be explained using this single expression, 1314 is the correct continuation of the series.

Option D:
Option D, 1320, exceeds the polynomial value by 6 and cannot be written as n⁴+3n for n = 6. Choosing 1320 would require altering the underlying algebraic relationship at the last term. Hence this option does not preserve the internal logic of the sequence and is incorrect.

This series is described by the rule aₙ = n⁴ + 5n² for n starting from 1. Evaluating for n = 1, 2, 3, 4 and 5 yields 1+5 = 6, 16+20 = 36, 81+45 = 126, 256+80 = 336 and 625+125 = 750. For n = 6 we obtain 6⁴+5·6² = 1296+180 = 1476. Therefore 1476 is the only term that keeps this quartic-plus-quadratic pattern intact for the next position.

Option A:
Option A, 1440, is lower than the formula’s output and does not equal n⁴+5n² for n = 6. It would require subtracting 36 from the correctly computed term only at the sixth step, which is not implied by the earlier values. This inconsistency makes option A incorrect.

Option B:
Option B, 1452, still fails to match 1296+180 and thus does not follow the same expression. Although numerically close, it cannot be produced by the rule for the sixth term. Hence option B would break the algebraic structure of the series and is not acceptable.

Option C:
Option C, 1464, lies between some candidate values but again does not satisfy aₙ = n⁴+5n² when n = 6. Selecting 1464 would mean slightly reducing the correct result without any justification from the preceding pattern. Therefore option C is not logically consistent with the sequence.

Option D:
Option D, 1476, coincides exactly with the value computed from the rule for n = 6. It preserves both the powers and coefficients that define the behaviour of the earlier terms. Because this option emerges naturally from the same formula, 1476 is the correct continuation of the number series.

This sequence follows the rule aₙ = n⁴ + 3n³ + n for n starting from 1. Evaluating for n = 1, 2, 3, 4 and 5 gives 1+3+1 = 5, 16+24+2 = 42, 81+81+3 = 165, 256+192+4 = 452 and 625+375+5 = 1005. For n = 6, we compute 6⁴+3·6³+6 = 1296+648+6 = 1950. Therefore 1950 is the unique next term that respects this quartic-plus-cubic-plus-linear pattern.

Option A:
Option A, 1950, matches exactly the result of n⁴+3n³+n when n = 6. It maintains the same combination of powers and coefficients that explain all the earlier terms. Because it arises directly from the established rule, this option correctly continues the sequence.

Option B:
Option B, 1938, is 12 less than the computed value and cannot be written as n⁴+3n³+n for n = 6. It implies a downward adjustment that does not appear anywhere in the earlier terms. Consequently, option B breaks the pattern and is incorrect.

Option C:
Option C, 1968, overshoots the correct polynomial value and likewise does not equal 6⁴+3·6³+6. Choosing 1968 would arbitrarily raise the sixth term relative to the rule. Therefore option C is not consistent with the functional relationship governing the series.

Option D:
Option D, 1980, departs even further from the formula’s output and has no basis in the given expression. Using 1980 would discard the tight connection between term position and term value, so option D cannot be correct.