nth term

Answer: option C, 29. The terms are consecutive perfect squares: 1 = 1², 4 = 2², 9 = 3², and 16 = 4². Therefore the nth term is aₙ = n². The 15th term is a₁₅ = 15² = 225, and the 14th term is a₁₄ = 14² = 196. Their difference is 225 - 196 = 29. A quicker method uses the difference of squares: 15² - 14² = (15 - 14)(15 + 14) = 1 × 29 = 29. Option A, 27, is too small; option B, 28, may come from an incorrect consecutive difference; option C, 29, follows exactly from the calculation; option D, 30, is one too large. In general, n² - (n-1)² = 2n - 1, so the difference ending at the 15th square is 2(15)-1 = 29. Memory cue: differences between consecutive squares are successive odd numbers.

The governing concept is the nth-term formula for an arithmetic progression. The first term is a=−5, and the common difference is d=4 because −1−(−5)=4, 3−(−1)=4, and 7−3=4. The formula is a_n=a+(n−1)d. Thus a_25=−5+(25−1)×4=−5+24×4=−5+96=91. Therefore option B is correct. The factor 24 appears because there are 24 equal gaps between the first term and the 25th term. Option A is four too small and corresponds to missing one increment. Options C and D add too many increments or mishandle the negative starting value. The constant difference confirms that this is arithmetic, not geometric.

The governing concept is evaluation of an explicit, or general, rule for a sequence. Substitute n = 7 and n = 5 separately into aₙ = n³ − n. We get a₇ = 7³ − 7 = 343 − 7 = 336, and a₅ = 5³ − 5 = 125 − 5 = 120. Therefore, a₇ − a₅ = 336 − 120 = 216, so option B is correct. A useful algebraic check is 7³ − 7 − (5³ − 5) = 343 − 7 − 125 + 5 = 216. The other options arise from an incorrect cube, forgetting the subtraction of n, or subtracting the terms incorrectly. Evaluating each term carefully avoids those errors.

The governing concept is finding terms from an explicit nth-term formula. The expression must be evaluated separately at the two requested indices, while preserving the parentheses. For n = 11, a_11 = 11(11 + 2) = 11 × 13 = 143. For n = 8, a_8 = 8(8 + 2) = 8 × 10 = 80. Subtracting the actual term values gives a_11 − a_8 = 143 − 80 = 63. Therefore option A is correct. A frequent mistake is to subtract the indices, or to interpret n(n + 2) as n² + 2; neither follows the stated formula. Options B, C and D are therefore not supported by the required substitution and subtraction.

The governing concept is using an explicit nth-term formula to find the position of a specified value. Set the formula equal to 148: n² + 5n − 2 = 148. Rearranging gives n² + 5n − 150 = 0. This factors as (n + 15)(n − 10) = 0, so the algebraic solutions are n = −15 and n = 10. A term number must be a positive integer, so n = −15 is not a valid sequence position. Direct substitution confirms the result: a_10 = 10² + 5(10) − 2 = 100 + 50 − 2 = 148. Therefore the 10th term is correct, which is option D. The 8th, 9th, and 11th terms do not equal 148.

The governing concept is identifying a term position from an explicit or general rule. We need an integer n such that n² + 2n = 168. Substituting n = 12 gives a_12 = 12² + 2(12) = 144 + 24 = 168, so 168 is the 12th term. Algebraically, the equation is n² + 2n − 168 = 0, which factors as (n + 14)(n − 12) = 0. The roots are n = −14 and n = 12; only the positive root can represent a sequence position. A nearby-term check also confirms the answer: a_11 = 121 + 22 = 143, whereas a_13 = 169 + 26 = 195. Hence option C is correct, and the other listed positions do not produce 168.

The governing concept is finding a term position from an nth-term formula. Set 4n² − 1 equal to 575: 4n² − 1 = 575. Adding 1 to both sides gives 4n² = 576, and dividing by 4 gives n² = 144. Therefore n = 12 or n = −12. Since sequence positions are positive integers, only n = 12 is valid. Substitution verifies the result: a_12 = 4(12²) − 1 = 4 × 144 − 1 = 576 − 1 = 575. Hence option C is correct. The neighboring positions demonstrate the distinction: a_11 = 4 × 121 − 1 = 483 and a_13 = 4 × 169 − 1 = 675, so neither can equal 575. The negative square root is rejected because it cannot be a term index.

The governing concept is finding the first term that satisfies a strict inequality. For positive n, the rule a_n = n² + 2n is increasing, so checking consecutive values around the boundary is sufficient. At n = 9, a_9 = 9² + 2(9) = 81 + 18 = 99, which is not greater than 100. At n = 10, a_10 = 10² + 2(10) = 100 + 20 = 120, which is greater than 100. Therefore the first qualifying term is the 10th term, so option B is correct. The word “greater” means strictly more than 100, not equal to or at least a later value. Although the 11th and 12th terms also exceed 100, they cannot be the first because the 10th term already satisfies the condition.

Answer: option A, 271. The terms are consecutive cubes: 1 = 1³, 8 = 2³, 27 = 3³, and 64 = 4³. Thus the general term is aₙ = n³. Hence a₉ = 9³ = 729 and a₁₀ = 10³ = 1000. Subtracting gives a₁₀ - a₉ = 1000 - 729 = 271. The difference-of-cubes identity provides a second check: 10³ - 9³ = (10-9)(10² + 10×9 + 9²) = 1(100 + 90 + 81) = 271. Option A is therefore correct. Option B, 273, results from an arithmetic error near 271. Option C, 275, may result from estimating rather than cubing exactly. Option D, 279, is also inconsistent with the exact subtraction. Be careful that a₁₀ means 10³, not 10×3. Memory cue: identify the position first, then apply the cube rule.

The governing concept is the nth term of a geometric progression. Each term is obtained by multiplying the previous term by 3: 18/6 = 3, 54/18 = 3, and 162/54 = 3. Thus the first term is a_1 = 6 and the common ratio is r = 3. The formula a_n = a_1r^(n−1) gives a_n = 6·3^(n−1) = 2·3^n, so option C is correct. Checking n = 1, 2, 3, and 4 gives 6, 18, 54, and 162. Option A is linear, option B gives 6, 12, 30, 84, and option D uses ratio 2, giving 6, 12, 24, 48. Only C has the required ratio and terms.

The governing concept is finding a quadratic nth-term rule by examining finite differences. The first differences are 5−1 = 4, 11−5 = 6, and 19−11 = 8, so they are not constant. The second differences are 6−4 = 2 and 8−6 = 2, indicating a quadratic expression. Test option C: for n = 1, a_1 = 1² + 1 − 1 = 1; for n = 2, a_2 = 4 + 2 − 1 = 5; for n = 3, a_3 = 9 + 3 − 1 = 11; and for n = 4, a_4 = 16 + 4 − 1 = 19. Hence C is correct. The other choices produce different values, so they cannot represent all four terms.

The governing concept is substituting term numbers into an explicit sequence rule and then subtracting. For n=10, a₁₀=10²+10+3=100+10+3=113. For n=9, a₉=9²+9+3=81+9+3=93. Therefore a₁₀−a₉=113−93=20, so option C is correct. The result can also be checked directly by simplifying: (10²+10+3)−(9²+9+3)=100+10−81−9=20, because the constant terms cancel. The values 18, 19, and 21 arise from arithmetic or substitution mistakes, such as using an incorrect square or omitting one of the linear terms. Calculating both terms carefully avoids those errors.

An explicit rule must reproduce every listed term from its position. The first differences are 17−5=12, 37−17=20, and 65−37=28. The second differences are 20−12=8 and 28−20=8, indicating a quadratic pattern. Test option B: for n=1, 4(1)²+1=5; for n=2, 4(2)²+1=17; for n=3, 4(3)²+1=37; and for n=4, 4(4)²+1=65. It matches all four terms, so B is correct. Option A is linear and gives 9 when n=2. Option C gives 5 at n=1 but 20 at n=2. Option D gives 5, 17, 29, 41, so it fails from the third term. Both direct substitution and the constant second difference support B.

The governing concept is identifying a pattern of consecutive perfect cubes and expressing the cube bases in terms of the position n. The terms can be rewritten as 64 = 4^3, 125 = 5^3, 216 = 6^3, and 343 = 7^3. Thus, when n = 1, 2, 3, and 4, the corresponding bases are 4, 5, 6, and 7. These bases follow the rule n + 3, so the general term is a_n = (n + 3)^3. Hence option B is correct. Option A gives (1 + 2)^3 = 27 as the first term. Option C gives 64 for n = 1 but gives 71 for n = 2, so it fails immediately afterward. Option D also does not produce the listed cube values. Checking more than one term confirms the rule rather than relying on a single match.

The governing concept is finding an explicit general term by recognizing perfect squares and then identifying the pattern in their bases. The terms are 9 = 3^2, 49 = 7^2, 121 = 11^2, and 225 = 15^2. The bases 3, 7, 11, 15 form an arithmetic sequence with first term 3 and common difference 4. Therefore, the nth base is 3 + (n − 1)4 = 4n − 1. Squaring this base gives a_n = (4n − 1)^2, so option A is correct. Verification gives 9 when n = 1 and 49 when n = 2. Option B produces 9, 25, 49, so its second term is wrong; options C and D also fail when tested against the second or third term. Matching several terms is essential.

The governing concept is the nth-term formula for an arithmetic sequence. The first term is a_1 = 15, and the common difference is d = 9 because 24 − 15 = 9 and 33 − 24 = 9. Hence a_n = a_1 + (n − 1)d = 15 + 9(n − 1) = 9n + 6. Substituting n = 40 gives a_40 = 9(40) + 6 = 360 + 6 = 366. The same result comes from noting that there are 39 intervals from the first term to the fortieth term: 15 + 39 × 9 = 15 + 351 = 366. Therefore, option B is correct. Option A is one difference too small, option C is one difference too large, and option D is two differences too large. The n − 1 factor is essential because the first term already has index 1.

The governing concept is the nth-term formula aₙ = a₁ + (n − 1)d. In this progression, a₁ = 3 and d = 8 − 3 = 5. For n = 10, a₁₀ = 3 + (10 − 1)×5 = 3 + 45 = 48. Therefore option D is correct. The subtraction of one from the term number is essential because the first term already has its starting value and no difference has yet been added; nine differences are needed to reach the tenth term. Option A results from adding only eight differences, option B from omitting the first term in the calculation, and option C does not satisfy the progression formula.

The governing concept is the general-term formula for an arithmetic progression: a_n = a_1 + (n − 1)d. The first term is a_1 = 12, and the common difference is d = 17 − 12 = 5. Substitution gives a_n = 12 + (n − 1)5 = 12 + 5n − 5 = 5n + 7. Hence option A is correct. Verification confirms the rule: at n = 1 it gives 5(1) + 7 = 12; at n = 2 it gives 17; and at n = 3 it gives 22. Option B has an incorrect coefficient and constant. Option C gives 17 when n = 1 instead of 12, while option D implies a common difference of 12 rather than 5.

The governing concept is the nth-term formula for an arithmetic progression: aₙ=a₁+(n−1)d. Here a₁=100 and d=90−100=−10 because every term decreases by 10. For the 11th term, ten equal differences occur after the first term. Therefore a₁₁=100+(11−1)(−10)=100−100=0. Hence option A is correct. Listing the terms confirms it: 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0. Options B, C, and D result from counting fewer than ten decreases or ignoring the negative sign in the common difference. The factor n−1 is essential because the first term itself is not obtained after one decrease.

An arithmetic progression is a sequence in which the same number is added each time. Here the first term is 13, and the common difference is 6 because 19 − 13 = 6. The nth-term rule is \\(a_n=a+(n-1)d\\), where a is the first term and d is the common difference. We must find the position of 61, not merely check whether it appears near the beginning.

Put the given value into the rule: \\(61=13+(n-1)6\\). Subtracting 13 gives \\(48=6(n-1)\\), so \\(n-1=8\\) and \\(n=9\\). Checking the sequence confirms this: the ninth term is 13 plus eight groups of 6, which equals 61. Therefore, option C, the 9th term, is correct. Options 7, 8, and 10 give different terms because they use the wrong number of common differences.

Use the arithmetic-progression rule a_n = a_1 + (n − 1)d. The first term is 11 and the common difference is d = 8 − 11 = −3. Continuing the sequence gives 11, 8, 5, 2, −1, −4, −7, ... . The first value less than zero is −1, and it occurs at n = 5. Therefore option B, the fifth term, is correct. The fourth term is 2, which is still positive, so it cannot answer the question. Although the sixth and seventh terms are also negative, they appear later and therefore are not the first negative term. The word “first” requires checking the preceding term as well as the negative one.

The governing concept is the nth-term formula for an arithmetic progression: a_n = a + (n − 1)d. Here the first term is a = 28 and the common difference is d = 24 − 28 = −4. For n = 9, a9 = 28 + (9 − 1)(−4) = 28 + 8(−4) = 28 − 32 = −4. Hence option A is correct. The negative answer is reasonable because the sequence decreases by 4 at every step: the terms are 28, 24, 20, 16, 12, 8, 4, 0, −4. Options B, C, and D result from using the wrong number of steps or stopping before the ninth term. The formula counts eight differences from the first to the ninth term.

The governing concept is the nth-term formula for an arithmetic progression: aₙ = a + (n − 1)d. The fifth term is given as a₅ = 27 and the common difference is d = 3. Substituting n = 5 gives 27 = a + (5 − 1)3 = a + 12. Therefore, a = 27 − 12 = 15, so option B is correct. Another way to see it is to move backward four equal steps from the fifth term: 27, 24, 21, 18, 15. Option A subtracts five differences instead of four, while options C and D subtract only three or two differences. Since the first term is four positions before the fifth term, exactly four common differences must be removed.

The governing concept is the nth-term formula for an arithmetic progression: aₙ = a₁ + (n − 1)d. The first term is a₁ = 18, and the common difference is d = 25 − 18 = 7. To locate 74, substitute it for aₙ: 74 = 18 + 7(n − 1). Subtracting 18 gives 56 = 7(n − 1), so n − 1 = 8 and n = 9. Therefore, 74 is the ninth term, making option C correct. A direct check confirms the result: the seventh term is 60, the eighth is 67, the ninth is 74, and the tenth is 81. Thus options A, B, and D identify neighboring positions rather than the required term.

The governing concept is the general, or nth, term of an arithmetic progression: a_n = a + (n − 1)d. The first term is a = 9 and the common difference is d = 17 − 9 = 8. Substituting these values gives a_n = 9 + (n − 1)8 = 9 + 8n − 8 = 8n + 1. Hence option A is correct. Checking n = 1 gives 9, n = 2 gives 17, and n = 3 gives 25, so the formula reproduces the listed sequence. Option B gives 17 for n = 1, option C also gives 17 for n = 1, and option D has a difference of only 1 between successive values. Therefore only option A has both the correct first term and the correct common difference.