nth term

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 nth-term formula for an arithmetic progression: aₙ = a + (n − 1)d, where a is the first term and d is the common difference. Here a = 9 and d = 17 − 9 = 8. Therefore, aₙ = 9 + (n − 1)8 = 9 + 8n − 8 = 8n + 1. Substituting n = 1 gives 9, and n = 2 gives 17, confirming option A. Option B gives 17 as its first value, option C gives 17 for n = 1, and option D represents a different pattern.

The governing concept is locating a term in an arithmetic progression. Use aₙ = a + (n − 1)d. The first term is a = 90 and the common difference is d = 82 − 90 = −8. Set the nth term equal to 18: 18 = 90 + (n − 1)(−8). Thus, 18 − 90 = −72 = −8(n − 1), so n − 1 = 9 and n = 10. Therefore, 18 is the tenth term, so option B is correct. The decreasing sign must be retained; ignoring it can lead to an incorrect position such as the ninth or eleventh term.

The governing concept is the nth-term rule for an arithmetic progression: aₙ = a₁ + (n − 1)d. The first term is a₁ = 2, and the common difference is d = 9 − 2 = 7. Therefore, a₈ = 2 + (8 − 1)×7 = 2 + 49 = 51, while a₃ = 2 + (3 − 1)×7 = 2 + 14 = 16. Adding the requested terms gives a₈ + a₃ = 51 + 16 = 67, so option A is correct. The factor n − 1 is essential because the first term is reached after zero common differences. Using n instead of n − 1, choosing the wrong indexed terms, or making an addition error can produce distractors such as 69, 71, or 73. The sequence itself confirms the calculation: its terms are 2, 9, 16, 23, 30, 37, 44, 51, so the eighth and third terms are indeed 51 and 16.

For an arithmetic progression, the nth term is found using \(a_n=a+(n-1)d\), where \(a\) is the first term and \(d\) is the common difference. In the given sequence, the first term is \(a=11\). The difference is \(18-11=7\), and the later differences also equal 7. We need the twelfth term, so use \(n=12\).

Substitution gives \(a_{12}=11+(12-1)7=11+11\times7=11+77=88\). Hence the twelfth term is 88, which is choice C. A common error is to multiply 12 by 7 without subtracting 1; that would count one extra difference. The first term itself uses zero differences, so the twelfth term uses exactly eleven differences.

The governing concept is using an explicit nth-term rule to find the position of a specified term. Set the given formula equal to 96: 7n − 2 = 96. Adding 2 to both sides gives 7n = 98, and dividing by 7 gives n = 14. Thus the fourteenth term is 96. Verification is immediate: a₁₄ = 7(14) − 2 = 98 − 2 = 96. The other options produce 75, 82, and 89 respectively when substituted into 7n − 2, so they cannot represent the required position. Therefore, option D is correct.

Use the nth-term formula aₙ = a₁ + (n − 1)d. Here a₁ = 50 and the common difference is d = 43 − 50 = −7, since the progression decreases by 7 each time. Set the nth term equal to 1: 50 + (n − 1)(−7) = 1. This gives 50 − 7(n − 1) = 1, so 7(n − 1) = 49, n − 1 = 7, and n = 8. Verification: the eighth term is 50 − 7 × 7 = 1. The other listed positions give different values, so option B is correct.

The governing concept is the nth-term formula for an arithmetic progression, a_n = a_1 + (n - 1)d. The first term is a_1 = 5, and the common difference is d = 14 - 5 = 9. Substitution gives a_n = 5 + (n - 1)9 = 5 + 9n - 9 = 9n - 4. Therefore option A is correct. A quick check confirms it: for n = 1, the expression gives 5; for n = 2, it gives 14; and for n = 3, it gives 23. Option B has the correct coefficient of n but the wrong constant, while options C and D do not preserve the common difference and fail the initial-term check.

The governing condition for a positive term is a_n > 0. Using the given rule, 5n - 12 > 0, so 5n > 12 and n > 12/5 = 2.4. The smallest positive integer index satisfying this is n = 3. Direct evaluation confirms the result: a_1 = -7, a_2 = -2, and a_3 = 15 - 12 = 3, which is positive. Therefore the first positive term is the 3rd term, so option B is correct. Option A is wrong because a_2 is negative; options C and D are also positive terms, but they occur later and therefore cannot be the first positive one.

The governing concept is the nth-term formula for an arithmetic progression: a_n = a_1 + (n - 1)d. In this sequence, a_1 = 13 and d = 18 - 13 = 5. For n = 20, a_20 = 13 + (20 - 1)5 = 13 + 95 = 108. Hence option B is correct. Another check is to note that the 20th term is reached after 19 equal increments from the first term, so adding 20 times the difference would incorrectly give 113. Option A corresponds to using only 18 increments, while options C and D arise from adding too many increments or mishandling the first term. The formula avoids these indexing errors.

The governing concept is the nth-term formula for an arithmetic progression: aₙ = a + (n − 1)d. Here the first term is a = 13, and the common difference is d = 19 − 13 = 6. For the fourteenth term, substitute n = 14: a₁₄ = 13 + (14 − 1) × 6 = 13 + 13 × 6 = 13 + 78 = 91. Therefore, option B is correct. Option A results from adding only twelve differences, while option C adds one difference too many; option D uses an incorrect difference or term count. The answer can also be checked by continuing the sequence: each successive term increases by 6, and the fourteenth term is 91.

The governing concept is the explicit or general rule of an arithmetic progression. The first term is a = 9 and the common difference is d = 15 − 9 = 6. Hence aₙ = a + (n − 1)d = 9 + 6n − 6 = 6n + 3. Substituting n = 1 gives 9, n = 2 gives 15, and n = 3 gives 21, so the rule reproduces the sequence. Therefore, option A is correct. Option B gives 9 for the first term but then increases by 9, not 6. Option C gives 3 as its first term, and option D gives 9 initially but has common difference 3. Checking both the first term and the difference removes these distractors.

This is an nth-term problem for an arithmetic progression. The first term is a = 64 and the common difference is d = 57 − 64 = −7. Using aₙ = a + (n − 1)d, set the term equal to 8: 8 = 64 + (n − 1)(−7). Hence 8 = 64 − 7(n − 1), so 7(n − 1) = 56, n − 1 = 8, and n = 9. Therefore, 8 is the ninth term, making option B correct. The negative difference shows that the progression decreases, but the same formula still applies. Options A, C, and D correspond to using too few or too many backward steps.

The governing formula for the nth term of an arithmetic progression is aₙ = a + (n − 1)d. Here the first term is a = 8, and the common difference is d = 17 − 8 = 9. Therefore, aₙ = 8 + (n − 1)9 = 8 + 9n − 9 = 9n − 1. Hence option A is correct. A quick verification is useful: when n = 1, the formula gives 9(1) − 1 = 8; when n = 2, it gives 17; and when n = 3, it gives 26. Option B does not produce the first term, option C has the wrong constant, and option D has the wrong common difference.

The nth-term formula for an arithmetic progression is aₙ = a + (n − 1)d. Here the first term is a = 17 and the common difference is d = 22 − 17 = 5. For n = 25, a₂₅ = 17 + (25 − 1)5 = 17 + 24 × 5 = 17 + 120 = 137. Therefore, option B is correct. The factor 24 is important because the first term requires zero jumps, so the 25th term requires 24 common differences. Option A may result from using only 23 differences, option C from adding one extra 5, and option D from another arithmetic error. Substituting n = 1 in the formula also returns 17, confirming that the indexing is correct.

The governing concept is the nth-term formula for an arithmetic progression: a_n = a_1 + (n - 1)d. Here the first term is a_1 = 12, and the common difference is d = 20 - 12 = 8; the same difference appears between every consecutive pair. Substituting these values gives a_n = 12 + (n - 1)8 = 12 + 8n - 8 = 8n + 4. Therefore option A is correct. A quick check confirms it: for n = 1, the formula gives 12; for n = 2 it gives 20; and for n = 4 it gives 36. Option C gives 4 at n = 1, option B gives 12n, and option D gives 12 at n = 1 but fails at later terms.

The governing concept is the nth-term structure of an arithmetic progression. The displayed terms begin with a_1 = x and have common difference d = 9. Therefore the fifth term is a_5 = a_1 + (5 - 1)d = x + 4×9 = x + 36. Since this term is given as 58, solve x + 36 = 58, which gives x = 22. Thus option B is correct. Directly listing the terms gives x, x+9, x+18, x+27, x+36, so the same result is clear. Options 20, 24, and 26 would make the fifth term 56, 60, and 62 respectively, not 58.

The governing concept is the nth-term formula for an arithmetic progression: a_n = a + (n - 1)d. Here the first term is a = 72 and the common difference is d = 64 - 72 = -8. Therefore, a_n = 72 + (n - 1)(-8). For the term to be zero, solve 72 - 8(n - 1) = 0. This gives 8(n - 1) = 72, so n - 1 = 9 and n = 10. Thus the zero appears as the tenth term, so option C is correct. Option A, B, and D result from using an incorrect number of decreases or miscounting the first term.

The governing concept is the general term of an arithmetic progression, a_n = a + (n - 1)d. The first term is a = 11, and the common difference is d = 20 - 11 = 9; the same difference is also seen from 29 - 20 and 38 - 29. Substitution gives a_n = 11 + (n - 1)9 = 11 + 9n - 9 = 9n + 2. Hence option A is correct. A quick check confirms it: when n = 1, 9(1) + 2 = 11, and when n = 2, the expression gives 20. Option C incorrectly keeps the first term as an added constant, while B and D use unsuitable coefficients.

The governing condition for a positive term is a_n > 0. Using the given rule, 7n - 23 > 0, so 7n > 23 and n > 23/7, which is approximately 3.29. The smallest positive integer position satisfying this is n = 4. Direct checking is also useful: a_3 = 7(3) - 23 = -2, which is not positive, whereas a_4 = 7(4) - 23 = 5, which is positive. Since the sequence increases by 7 each time, every later term is also positive, so the first positive term is the fourth term. Therefore option B is correct; options C and D identify later positive terms, not the first one.