UGC NET Questions (Paper – 1)

The pattern here arises from the formula aₙ = 2n⁴−1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 2·1⁴−1 = 1, 2·2⁴−1 = 31, 2·3⁴−1 = 161, 2·4⁴−1 = 511 and 2·5⁴−1 = 1249. For n = 6 we compute 2·6⁴−1 = 2·1296−1 = 2592−1 = 2591. Therefore 2591 is the only term that correctly extends the quartic-based pattern.

Option A:
Option A, 2511, is far below the calculated value and does not equal 2n⁴−1 for any n that continues the sequence. It ignores the rapid growth implied by the quartic expression. Thus 2511 is not a valid next term.

Option B:
Option B, 2541, is still different from 2591 and cannot be obtained from 2·6⁴−1. Choosing it would break the exact algebraic rule that explains all the previous terms. Hence 2541 cannot be the correct answer.

Option C:
Option C, 2591, matches the result of evaluating 2n⁴−1 when n = 6. It preserves the quartic dependence on n and continues the rapid increase seen in the sequence. For these reasons, 2591 is the correct continuation.

Option D:
Option D, 2651, overshoots the correct value and is not equal to 2n⁴−1 for any appropriate n in order. Adopting 2651 would distort the pattern and introduce an unjustified jump. Therefore it is not a valid option.

This series is generated by the expression (a_n = 2n^4 - n + 3) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2−1+3 = 4, 32−2+3 = 33, 162−3+3 = 162, 512−4+3 = 511 and 1250−5+3 = 1248. For n = 6 we compute (2×6^4 - 6 + 3 = 2592 - 3 = 2589). Therefore 2589 is the correct next term.

Option A:
Option A, 2589, coincides with the value of (2n^4 - n + 3) when n = 6. It maintains the same combination of quartic and linear terms that explain all previous numbers. Because it fits the rule exactly, 2589 is the correct continuation.

Option B:
Option B, 2557, is 32 less than the computed value and does not satisfy the expression for n = 6. It would require a sudden drop that has no basis in the pattern, so 2557 is not valid.

Option C:
Option C, 2613, overshoots the predicted value and cannot be obtained from the generating rule. Selecting 2613 would arbitrarily increase the last term and break the algebraic structure. Thus 2613 is not correct.

Option D:
Option D, 2645, deviates even further from 2589 and is not the output of (2n^4 - n + 3) for any integer n continuing the sequence. Adopting 2645 would disrupt the pattern, so it is not the right answer.

This sequence is defined by the rule (a_n = 3n^4 - n^2 + 2) with n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 3−1+2 = 4, 48−4+2 = 46, 243−9+2 = 236, 768−16+2 = 754 and 1875−25+2 = 1852. For n = 6 we calculate (3×6^4 - 6^2 + 2 = 3888 - 36 + 2 = 3854). Therefore 3854 is the correct next term.

Option A:
Option A, 3854, is exactly the value produced by the formula for n = 6. It preserves the quartic and quadratic contributions in the same way as for the earlier terms and keeps the pattern coherent. Thus 3854 is the correct continuation of the series.

Option B:
Option B, 3818, is 36 less than the correct value and does not equal (3×6^4 - 6^2 + 2). It suggests an unwarranted reduction that is not reflected in the previous behaviour. Hence 3818 is not valid.

Option C:
Option C, 3886, overshoots the computed value and again fails to satisfy the formula. Choosing 3886 would break the algebraic structure and introduce an arbitrary jump. Therefore 3886 is not correct.

Option D:
Option D, 3922, deviates even more from 3854 and has no basis in the generating expression. Using 3922 would destroy the close match between the rule and the data, so it is not the right answer.

The terms are generated by aₙ = 4n⁴ − n² + 3 with n starting at 1. Substituting n = 1, 2, 3, 4 and 5 gives 6, 63, 318, 1011 and 2478, which matches the series. For n = 6 we obtain a₆ = 4·6⁴ − 6² + 3 = 4·1296 − 36 + 3 = 5184 − 33 = 5151. Thus 5151 is the only term consistent with this quartic pattern.

Option A:
Option A, 5127, is 24 less than the value predicted by 4n⁴ − n² + 3 at n = 6. Adopting 5127 would mean arbitrarily reducing the correct polynomial result at the final step. Since earlier values fit the rule exactly, this modification is unjustified, so option A is incorrect.

Option B:
Option B, 5139, is 12 less than the correct value 5151 and cannot be obtained from 4·6⁴ − 6² + 3. It suggests a slight downward deviation that does not follow from the formula. Therefore this option fails to provide a mathematically sound continuation of the series.

Option C:
Option C, 5151, is precisely the number produced when n = 6 is substituted into 4n⁴ − n² + 3. It maintains the same balance between the quartic term and the subtractive quadratic component that governs the earlier terms. Because it extends the pattern without any alteration, this option is correct.

Option D:
Option D, 5163, is 12 greater than the polynomial value and likewise cannot be generated by the rule at n = 6. Introducing such an increase at only one term would break the consistent algebraic structure. Hence option D is not a valid continuation.

The series satisfies the rule (a_n = 2n^4 - 3n + 5) with n starting from 1. For n = 1, 2, 3, 4 and 5 we get 2−3+5 = 4, 32−6+5 = 31, 162−9+5 = 158, 512−12+5 = 505 and 1250−15+5 = 1240. For n = 6 we compute (2×6^4 - 18 + 5 = 2592 - 13 = 2579). Hence 2579 is the correct next term.

Option A:
Option A, 2543, is 36 less than the value predicted by the rule and does not equal (2×6^4 - 3×6 + 5). It introduces an arbitrary shortfall not reflected earlier, so 2543 is not valid.

Option B:
Option B, 2561, is closer but still fails to match 2579 and cannot be derived from the formula for n = 6. Choosing 2561 would alter the established pattern. Thus 2561 is not correct.

Option C:
Option C, 2597, overshoots the computed value and again does not satisfy the expression for the next index. Using 2597 would inflate the term without justification and break the quartic structure. Hence 2597 is not appropriate.

Option D:
Option D, 2579, equals the exact output of (a_n = 2n^4 - 3n + 5) when n = 6. It maintains the same relationship between index and term for all entries in the series and is therefore the correct continuation.

Each term of the series can be written as n⁴+1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 1⁴+1 = 2, 2⁴+1 = 17, 3⁴+1 = 82, 4⁴+1 = 257 and 5⁴+1 = 626. The next term corresponds to n = 6, which gives 6⁴+1 = 1296+1 = 1297. Therefore 1297 is the only value that continues the same quartic pattern.

Option A:
Option A matches the value obtained from the rule aₙ = n⁴+1 when n = 6. It extends the exact same relationship that holds for all the earlier terms of the sequence. Because no change in the functional rule is needed to reach 1297, this option is fully consistent with the observed pattern.

Option B:
Option B, 1257, is close in magnitude but does not equal 6⁴+1 and therefore fails to satisfy the quartic formula for n = 6. Choosing it would require altering the rule that clearly generates the previous terms. Hence it cannot represent the correct continuation of the series.

Option C:
Option C, 1273, similarly does not arise from the expression n⁴+1 for any integer n that follows the existing index order. It breaks the tight link between term position and value that is evident in the given numbers. Thus 1273 is not a valid next term.

Option D:
Option D, 1327, overshoots the correct quartic value and cannot be expressed as 6⁴+1. Adopting 1327 would mean abandoning the neat polynomial structure and introducing an arbitrary adjustment. Therefore it is not the correct answer.

The series can be expressed by the formula aₙ = (n⁴+n²)/2 for n starting from 1. For n = 1, 2, 3, 4 and 5 we get (1+1)/2 = 1, (16+4)/2 = 10, (81+9)/2 = 45, (256+16)/2 = 136 and (625+25)/2 = 325, matching the terms given. For n = 6 the expression gives (6⁴+6²)/2 = (1296+36)/2 = 1332/2 = 666. Thus 666 is the unique next term that fits this quartic-based sequence.

Option A:
Option A, 646, is 20 less than the computed value and does not equal (6⁴+6²)/2. It has no basis in the formula that successfully models all the earlier terms. Therefore 646 cannot serve as the correct continuation.

Option B:
Option B, 654, is still short of 666 and similarly fails to arise from the expression involving n⁴ and n². Choosing 654 would force a change in the pattern for the last term only, which is mathematically unsound. Hence it is not the right answer.

Option C:
Option C, 681, is larger than the predicted value and again does not match (6⁴+6²)/2. It represents an arbitrary modification rather than a logical extension of the rule. Thus 681 is not a valid next term.

Option D:
Option D, 666, exactly matches the output of the formula when n = 6. It preserves the structural relationship between the terms and their positions and keeps the series fully consistent. For this reason, 666 is the correct option.

This series is described by the rule (a_n = 2n^4 - n^2) for n starting from 1. For n = 1, 2, 3, 4 and 5 we get 2−1 = 1, 32−4 = 28, 162−9 = 153, 512−16 = 496 and 1250−25 = 1225, which matches the given list. For n = 6 we compute (2×6^4 - 6^2 = 2×1296 - 36 = 2592 - 36 = 2556). Therefore 2556 is the correct continuation of the quartic pattern.

Option A:
Option A, 2528, is 28 less than the formula’s output and does not equal (2×6^4 - 6^2). It would force an arbitrary downward adjustment at the last term, which is inconsistent with the rule. Hence 2528 is not a valid next term.

Option B:
Option B, 2556, matches exactly the value produced by (a_n = 2n^4 - n^2) when n = 6. It maintains the same combination of quartic and quadratic parts seen in all earlier terms. For this reason, 2556 is the correct answer.

Option C:
Option C, 2592, corresponds to (2×6^4) alone and ignores the subtraction of (6^2). That omission breaks the established formula. Thus 2592 does not correctly extend the sequence.

Option D:
Option D, 2616, is even larger and cannot be written as (2n^4 - n^2) for the next integer n. Adopting 2616 would make the pattern inconsistent, so it is not correct.

The sequence follows the formula (a_n = n^4 + 2n + 3) with n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 1+2+3 = 6, 16+4+3 = 23, 81+6+3 = 90, 256+8+3 = 267 and 625+10+3 = 638. For n = 6 the expression yields (6^4 + 12 + 3 = 1296 + 15 = 1311). Therefore 1311 is the correct next term under this quartic-plus-linear pattern.

Option A:
Option A, 1287, is 24 less than the calculated value and does not equal (6^4 + 2×6 + 3). It would require weakening the rule only at the final term, which is inconsistent. Hence 1287 is not valid.

Option B:
Option B, 1299, is still below 1311 and again cannot be written as (n^4 + 2n + 3) for n = 6. Adopting 1299 would disrupt the tight connection between index and term. Thus 1299 is not correct.

Option C:
Option C, 1311, matches exactly the output of the formula when n = 6. It preserves the same quartic and linear structure that generated all earlier values. For this reason, 1311 is the correct continuation.

Option D:
Option D, 1327, overshoots the correct value and fails to satisfy the expression for n = 6. Using 1327 would abandon the algebraic coherence of the series, so it is not a valid answer.

The sequence is described by the formula aₙ = n⁴−n+2 for n starting from 1. For n = 1, 2, 3, 4 and 5, we get 1−1+2 = 2, 16−2+2 = 16, 81−3+2 = 80, 256−4+2 = 254 and 625−5+2 = 622. For n = 6 the formula gives 1296−6+2 = 1292. Thus 1292 is the correct next term in line with this quartic-minus-linear rule.

Option A:
Option A, 1252, is 40 less than the computed value and does not equal n⁴−n+2 for n = 6. It breaks the exact pattern that has held for the earlier terms. Therefore 1252 is not a valid continuation.

Option B:
Option B, 1268, is still lower than 1292 and likewise fails to come from the generating expression. Selecting 1268 would require changing the formula without any clear reason. Hence it is not correct.

Option C:
Option C, 1292, matches perfectly with 6⁴−6+2 and continues the same structural relationship between n and aₙ. It maintains both the power and linear components consistently, making 1292 the correct answer.

Option D:
Option D, 1322, overshoots the correct value and cannot be obtained by substituting n = 6 into the rule. Using 1322 would distort the pattern and undermine the algebraic coherence of the sequence. Thus it is not appropriate.

This sequence is given by the formula (a_n = n^4 + n^3 + n) for n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 1+1+1 = 3, 16+8+2 = 26, 81+27+3 = 111, 256+64+4 = 324 and 625+125+5 = 755. For n = 6 we obtain (6^4 + 6^3 + 6 = 1296 + 216 + 6 = 1518). Therefore 1518 is the correct next term.

Option A:
Option A, 1494, is smaller than the formula’s output and does not equal (6^4 + 6^3 + 6). It would impose an unjustified reduction at the last step, so 1494 is not a valid continuation.

Option B:
Option B, 1506, is still below 1518 and fails to satisfy the expression for n = 6. Choosing 1506 would break the quartic-plus-cubic structure of the sequence. Thus 1506 is not correct.

Option C:
Option C, 1518, matches exactly the value produced by the rule when n = 6. It preserves the same combination of quartic and cubic terms that generated all the earlier numbers and hence is the correct next term.

Option D:
Option D, 1530, overshoots the computed value and cannot be written as (n^4 + n^3 + n) for the next index. Using 1530 would distort the carefully balanced pattern, so it is not appropriate.

The sequence is generated by the rule aₙ = n⁴−n²+2 for n starting from 1. For n = 1, 2, 3, 4 and 5 we compute 1−1+2 = 2, 16−4+2 = 14, 81−9+2 = 74, 256−16+2 = 242 and 625−25+2 = 602. For n = 6 the expression gives 1296−36+2 = 1262. Hence 1262 is the term that preserves this quartic-minus-square pattern.

Option A:
Option A, 1222, is 40 less than the value predicted by n⁴−n²+2 for n = 6. It does not satisfy the formula and would require adjusting the result in a way that is not supported by the earlier data. Therefore 1222 is not the correct continuation.

Option B:
Option B, 1238, also fails to match 1296−36+2 and breaks the tight link between index and term. It represents a partial correction but still does not arise from the generating rule. Thus 1238 cannot be accepted.

Option C:
Option C, 1292, overshoots the correct value and cannot be written as n⁴−n²+2 for n = 6. Adopting 1292 would abandon the exact quartic pattern that fits the sequence so well. Hence it is not valid.

Option D:
Option D, 1262, exactly equals 6⁴−6²+2 and therefore continues the same polynomial relationship. It keeps the structure of the series fully consistent from the first term to the next one. For this reason, 1262 is the correct next term.

The series can be generated by the formula (a_n = 2n^4 + 3n^2 + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 6, 45, 190, 561 and 1326, which match the question. For n = 6 the expression gives (2×6^4 + 3×6^2 + 1 = 2592 + 108 + 1 = 2701). Hence 2701 is the correct next term.

Option A:
Option A, 2665, is 36 less than the formula’s output and does not equal (2×6^4 + 3×6^2 + 1). It would require an unjustified reduction at the final term, so 2665 is not valid.

Option B:
Option B, 2683, remains below 2701 and fails to satisfy the expression for n = 6. Choosing 2683 would disrupt the quartic-plus-square structure of the sequence. Thus 2683 is not correct.

Option C:
Option C, 2723, overshoots the computed value and again cannot be obtained from the given formula. Using 2723 would break the algebraic pattern that has held so far. Hence 2723 is not appropriate.

Option D:
Option D, 2701, exactly equals the value generated by the rule for n = 6. It maintains the same functional relationship between n and a_n for all terms, so 2701 is the correct continuation of the series.

The terms follow the rule aₙ = 2n⁴ + 2n³ + n with n starting from 1. For n = 1, 2, 3, 4 and 5 this formula produces 5, 50, 219, 644 and 1505, matching the given series perfectly. Evaluating the same expression for n = 6 yields 3030. Hence 3030 is the only number that preserves this quartic-plus-cubic pattern.

Option A:
Option A, 3006, is 24 less than the value obtained from aₙ = 2n⁴ + 2n³ + n at n = 6. To accept 3006 we would need to reduce the polynomial result arbitrarily at the sixth term. This change is not reflected anywhere else in the progression, so option A is incorrect.

Option B:
Option B, 3020, is 10 less than the correct value and also fails to equal the formula’s output for n = 6. Although numerically close, it breaks the precise algebraic relationship between n and aₙ. Therefore option B does not correctly continue the series.

Option C:
Option C, 3030, matches exactly the value produced by the expression when n = 6. It maintains the same contributions from the fourth-power and cubic terms along with the linear part as in earlier terms. Because this rule explains the entire sequence and extends naturally to 3030, option C is correct.

Option D:
Option D, 3046, is 16 greater than the expected value and cannot be derived from the formula at n = 6. Choosing 3046 would introduce an unjustified increase at the last step, disrupting the pattern. Thus option D is not a valid continuation.

The series follows the rule (a_n = n^4 + 4n^2 - 2) with n starting from 1. For n = 1, 2, 3, 4 and 5 the expression yields 3, 30, 115, 318 and 723, as in the question. For n = 6 we compute (6^4 + 4×6^2 - 2 = 1296 + 144 - 2 = 1438). Therefore 1438 is the correct next term in the sequence.

Option A:
Option A, 1414, is 24 less than the formula’s output and does not equal (6^4 + 4×6^2 - 2). It would impose an artificial reduction that the pattern does not support. Hence 1414 is not consistent with the rule.

Option B:
Option B, 1438, matches exactly the result of the expression for n = 6. It preserves the quartic and quadratic contributions in exactly the same way as for earlier terms. This alignment makes 1438 the correct answer.

Option C:
Option C, 1454, overshoots the computed value and cannot be derived from the formula for n = 6. Choosing 1454 would disrupt the precise algebraic structure of the series. Thus 1454 is not valid.

Option D:
Option D, 1470, deviates even more from 1438 and once again fails to satisfy the rule. Using 1470 would break the tight connection between index and term, so it is not correct.