UGC NET Questions (Paper – 1)

The series is described by the quadratic formula aₙ = 3n²+2n+1 for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 6, 17, 34, 57 and 86, which match the given terms exactly. For n = 6 we obtain 3·6²+2·6+1 = 108+12+1 = 121. Hence the next term must be 121 to preserve the same quadratic relationship.

Option A:
Option A, 111, does not equal 3·6²+2·6+1 and therefore fails to extend the polynomial rule consistently. It would require lowering the value obtained from the formula by 10, which has no justification in the structure of the sequence. Thus 111 cannot be the correct continuation.

Option B:
Option B, 121, is exactly the value produced by the formula aₙ = 3n²+2n+1 when n = 6. It maintains the same dependence on n that explains all earlier terms. Because the pattern holds perfectly when this value is used, 121 is the correct next term.

Option C:
Option C, 115, lies between 111 and 121 but does not arise from substituting n = 6 into the quadratic expression. Selecting 115 would break the neat quadratic pattern and make the rule inconsistent. Therefore it is not the right answer.

Option D:
Option D, 129, is larger than the value predicted by the formula and again cannot be written as 3n²+2n+1 for n = 6. Using 129 would lose the tight algebraic link between n and aₙ, so it is not a valid continuation.

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.

The series follows the rule (a_n = n^3 + 2) for n starting from 1. Substituting n = 1, 2, 3, 4 and 5 gives 3, 10, 29, 66 and 127, which match the given terms. For n = 6 the same formula yields (6^3 + 2 = 216 + 2 = 218). Therefore 218 is the unique value that continues the same cubic pattern without any change.

Option A:
Option A equals 218, which is exactly the value of (n^3 + 2) when n = 6. It preserves the single algebraic rule that generates all previous terms. Because no adjustment or exception is needed to obtain 218, this option correctly extends the series.

Option B:
Option B, 214, does not equal (6^3 + 2) and would require subtracting an additional 4 from the formula’s output. This breaks the clean relationship between each term and its index. Hence 214 cannot be accepted as the correct next term.

Option C:
Option C, 222, is larger than the formula’s value and does not arise from (n^3 + 2) for the next integer. Choosing 222 would arbitrarily increase the term and destroy the exact cubic correspondence. Therefore 222 is not a valid continuation of the series.

Option D:
Option D, 226, deviates even further from 218 and again fails to satisfy the expression (n^3 + 2) for n = 6. It would imply altering the rule only at the last step, which is mathematically inconsistent. Thus 226 is not correct.

The series follows the formula aₙ = n³+2n² for n starting from 1. For n = 1, 2, 3, 4 and 5 we calculate 1+2 = 3, 8+8 = 16, 27+18 = 45, 64+32 = 96 and 125+50 = 175. For n = 6 the expression gives 216+72 = 288. Hence 288 is the term that correctly extends this cubic-plus-square pattern.

Option A:
Option A, 278, is 10 less than the required value and does not equal n³+2n² for n = 6. It breaks the exact numeric relationship that has generated all previous terms. Therefore 278 is not the correct continuation.

Option B:
Option B, 282, is closer but still not equal to 216+72. It would imply a small but unjustified alteration of the formula at the last step. Thus 282 does not fit the pattern.

Option C:
Option C, 296, exceeds the computed value and again does not come from n³+2n² when n = 6. Choosing 296 would destroy the perfectly consistent algebraic structure of the sequence. Hence it is not valid.

Option D:
Option D, 288, is exactly 216+72 and comes directly from the rule aₙ = n³+2n². It preserves both the cubic and quadratic contributions in the same way as before, making 288 the correct next term.

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 can be described by the formula (a_n = n^3 + 2) for n starting from 2. For n=2,3,4,5 and 6 the values are 8+2=10, 27+2=29, 64+2=66, 125+2=127 and 216+2=218, matching the given terms. The next index is n=7, giving (7^3+2 = 343+2 = 345). Thus 345 is the natural continuation of this cubic-based sequence.

Option A:
Option A, 337, does not arise from the expression (n^3+2) for any integer n that continues the index sequence. Using 337 would require abandoning the clear functional rule that explains all earlier terms. Therefore 337 is not a valid next term.

Option B:
Option B equals 345, which is exactly (7^3+2), following the same formula used to generate 10, 29, 66, 127 and 218. Because the rule holds for every term including the new one, 345 is the correct answer.

Option C:
Option C, 349, is larger than the value predicted by the cubic formula and cannot be justified without changing the rule. It does not match (n^3+2) for any integer n that logically follows. Hence 349 is not consistent with the pattern.

Option D:
Option D, 353, similarly fails to satisfy the relation (a_n = n^3+2) for the next integer in sequence. Adopting 353 would make the series lose its simple and elegant algebraic structure. Thus 353 is not the right continuation.

This series is given by the rule (a_n = 2n^3 + 3n^2 + 1) for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 6, 29, 82, 177 and 326, which match the listed terms. For n = 6 we compute (2×6^3 + 3×6^2 + 1 = 432 + 108 + 1 = 541). Thus 541 is the unique next term that maintains this cubic-plus-square pattern.

Option A:
Option A, 529, is 12 less than the formula’s result and does not equal (2×6^3 + 3×6^2 + 1). It would imply altering the rule at the last term, which is not supported by the sequence. Therefore 529 is not correct.

Option B:
Option B, 541, coincides exactly with the value produced by the expression for n = 6. It preserves all the coefficients and the constant term used for earlier values. For this reason, 541 is the correct continuation.

Option C:
Option C, 547, overshoots the computed value and cannot be written as (2n^3 + 3n^2 + 1) for the next index. Choosing 547 would break the algebraic consistency, so it is not valid.

Option D:
Option D, 553, deviates even more from the formula and again fails to satisfy the same relationship for n = 6. Adopting 553 would distort the pattern, so it is not the right answer.

The terms follow the rule aₙ = 2n⁴ + n³ + 2 with n starting at 1. Substituting n = 1, 2, 3, 4 and 5 yields 5, 42, 191, 578 and 1377, exactly the five numbers in the series. Evaluating the same expression for n = 6 gives 2810. Hence 2810 is the only value that fits this established polynomial pattern.

Option A:
Option A, 2786, is 24 less than the value obtained from aₙ = 2n⁴ + n³ + 2 for n = 6. To choose 2786 we would need to reduce the polynomial result only at the last term. Since earlier entries follow the rule without such a change, 2786 cannot correctly continue the series.

Option B:
Option B, 2798, is still below the required value and does not equal the output of the formula for n = 6. It represents a partial adjustment towards 2810 but still breaks the algebraic relationship. Therefore this option does not preserve the exact pattern linking the terms.

Option C:
Option C, 2810, matches precisely the value produced by 2n⁴ + n³ + 2 when n = 6. It maintains the contributions of both the fourth-power and cube terms in exactly the same way as for previous indices. Because the rule explains the given terms and extends naturally to 2810 next, this option is correct.

Option D:
Option D, 2822, is 12 larger than the polynomial value and cannot be generated by the same expression at n = 6. Accepting 2822 would require adding an unexplained increment at the final step. This would break the smooth growth dictated by the formula, so 2822 is not acceptable.

The pattern here is aₙ = 2n⁵ − 3n² + 4 with n beginning at 1. For n = 1, 2, 3, 4 and 5 this expression gives 3, 56, 463, 2004 and 6179, exactly reproducing the given sequence. When n = 6 we get a₆ = 2·6⁵ − 3·6² + 4 = 2·7776 − 108 + 4 = 15552 − 104 = 15448. Thus 15448 is the unique value consistent with this quintic pattern.

Option A:
Option A, 15448, is exactly the outcome of substituting n = 6 into 2n⁵ − 3n² + 4. It maintains the same dominance of the fifth-power term with a negative quadratic adjustment and a constant shift. Because this structure accounts for all earlier terms and extends naturally to 15448, this option correctly continues the series.

Option B:
Option B, 15424, is 24 less than the rule’s value and cannot be generated by the same polynomial at n = 6. Accepting 15424 would require reducing the correct result arbitrarily at the final term. This breaks the algebraic coherence of the sequence, so option B is incorrect.

Option C:
Option C, 15436, is 12 less than 15448 and again does not equal 2·6⁵ − 3·6² + 4. It represents an approximate but inaccurate candidate that does not come from the established rule. Therefore option C is not a valid next term.

Option D:
Option D, 15460, is 12 greater than the computed value and would demand increasing the polynomial output only at this step. Since earlier terms exactly match the expression, this modification is unjustified. Hence option D does not correctly extend the pattern.

This series is based on the formula aₙ = 2n³+1 for n starting from 1. For n = 1, 2, 3, 4 and 5 we obtain 2·1³+1 = 3, 2·2³+1 = 17, 2·3³+1 = 55, 2·4³+1 = 129 and 2·5³+1 = 251. For n = 6 the expression gives 2·6³+1 = 432+1 = 433. Therefore 433 is the next term that keeps the cubic pattern intact.

Option A:
Option A, 413, is 20 less than the value provided by 2·6³+1. It does not satisfy the generating formula and would require an unexplained downward adjustment. Hence 413 cannot be considered the correct continuation of the series.

Option B:
Option B, 421, is closer but still does not match 2·6³+1, so it fails to maintain the exact relationship between n and aₙ. Choosing it would make the pattern inconsistent with the earlier terms. Thus 421 is not a valid answer.

Option C:
Option C, 433, is exactly the value given by the rule aₙ = 2n³+1 when n = 6. It carries the same structure forward without alteration and fits perfectly with all preceding data. For this reason, 433 is the correct next term.

Option D:
Option D, 448, is larger than the formula’s output and cannot be obtained from 2n³+1 for n = 6. Adopting 448 would abandon the precise cubic relationship that defines the sequence, so it is not correct.

This series is generated by the formula aₙ = 5n³+n² for n starting from 1. For n = 1, 2, 3, 4 and 5 we have 5·1³+1 = 6, 5·2³+4 = 44, 5·3³+9 = 144, 5·4³+16 = 336 and 5·5³+25 = 650. For n = 6 the same expression yields 5·6³+36 = 1080+36 = 1116. Therefore 1116 is the correct next term that preserves this cubic-plus-square relationship.

Option A:
Option A, 1076, is 40 less than the computed value and does not satisfy aₙ = 5n³+n² for n = 6. It weakens the consistency between the algebraic rule and the numeric sequence. Hence 1076 is not the correct continuation.

Option B:
Option B, 1092, is somewhat closer but still different from 1116 and cannot be obtained from the formula without changing coefficients. This would make the pattern inconsistent at the final term. Thus 1092 is not valid.

Option C:
Option C, 1146, is larger than the correct value and again fails to equal 5n³+n² for n = 6. Choosing 1146 would distort the functional relationship that holds throughout the sequence. Therefore it is not the right answer.

Option D:
Option D, 1116, exactly matches the result given by the rule when n = 6. It keeps the cubic and quadratic contributions aligned in the same way as for earlier terms and thus is the correct next term.

The terms of this sequence are generated by aₙ = n⁵ + n³ + 2 with n beginning at 1. Substituting n = 1, 2, 3, 4 and 5 gives 4, 42, 272, 1090 and 3252, which agrees perfectly with the series. For n = 6 we obtain a₆ = 6⁵ + 6³ + 2 = 7776 + 216 + 2 = 7994. Hence 7994 is the unique value that preserves this quintic-plus-cubic pattern.

Option A:
Option A, 7970, is 24 less than the formula’s output at n = 6. To accept 7970 we would have to subtract 24 from the correct result only at the sixth term, even though earlier values match the formula exactly. This breaks the algebraic consistency of the series, so option A is incorrect.

Option B:
Option B, 7982, is 12 smaller than the correct value 7994 and cannot be produced by n⁵ + n³ + 2 for n = 6. It represents a near miss rather than a true continuation of the rule. Since the pattern is purely functional in n, any such deviation must be rejected.

Option C:
Option C, 7994, is exactly the result of substituting n = 6 into the expression n⁵ + n³ + 2. It maintains the same structure in which the fifth-power term dominates and the cubic and constant terms supply additional increments. Because it fits all given data and extends the rule without modification, this option is correct.

Option D:
Option D, 8006, is 12 greater than the computed value and again cannot arise from the same polynomial at n = 6. Accepting 8006 would introduce an unjustified extra amount at the final step. Therefore this option does not correctly extend the number series.

The pattern is captured by the polynomial aₙ = n⁵ − 2n² + 4 with n starting from 1. For n = 1, 2, 3, 4 and 5 this formula yields 3, 28, 229, 996 and 3079, exactly the given sequence. Evaluating the same expression at n = 6 gives 7708. Thus 7708 is the only value that continues this quintic pattern consistently.

Option A:
Option A, 7708, is precisely the value obtained from aₙ = n⁵ − 2n² + 4 when n = 6. It maintains the strong fifth-power growth combined with the quadratic subtraction and constant term that shape the earlier numbers. Because this rule explains all five terms and leads directly to 7708 next, this option is correct.

Option B:
Option B, 7684, is 24 less than the required value and does not equal the formula’s output for n = 6. To accept 7684 we would need to diminish the polynomial result only at this index. Such a modification is not supported by the rest of the sequence, so option B is incorrect.

Option C:
Option C, 7724, is 16 greater than the pattern value and again cannot be produced by n⁵ − 2n² + 4 at n = 6. Using 7724 would inflate the term beyond the rule’s prediction, breaking the algebraic structure. Therefore option C is not a valid continuation.

Option D:
Option D, 7740, deviates even more from 7708 and likewise fails to satisfy the expression at n = 6. Adopting 7740 would ignore the exact polynomial relationship linking the terms. Hence option D is not acceptable.

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.

The series can be generated using the expression 2n² + 1 for consecutive integers n. For n = 1 to 5 we obtain 2(1)² + 1 = 3, 2(2)² + 1 = 9, 2(3)² + 1 = 19, 2(4)² + 1 = 33 and 2(5)² + 1 = 51. The next integer is 6, so the next term should be 2(6)² + 1. This equals 2 × 36 + 1 = 72 + 1 = 73, which is consistent with the pattern.

Option A:
Option A gives 73, which exactly matches the formula 2n² + 1 when n = 6. The extended sequence 3, 9, 19, 33, 51, 73 is generated smoothly from one algebraic rule. This confirms that 73 is the correct continuation of the number series.

Option B:
Option B suggests 71, which does not satisfy the expression 2n² + 1 for any integer following 5 in this context. Using 71 would break the simple polynomial structure underlying the series. Therefore, 71 cannot be the correct next term.

Option C:
Option C provides 75, which also fails to match 2n² + 1 for the next integer. It introduces an inconsistent jump that is not supported by the existing values. Hence, 75 is not in harmony with the pattern of the sequence.

Option D:
Option D offers 77, which similarly does not correspond to 2n² + 1 for n = 6 or any immediate extension. Including 77 would abandon the clear quadratic rule. Thus, 77 is not the appropriate choice.