nth term

Observe the numerator and denominator patterns separately. The numerators are 1, 2, 3, 4, so the numerator of the nth term is n. The denominators are 2, 3, 4, 5, each one greater than its corresponding numerator, so the denominator is n + 1. Therefore the nth term is n/(n + 1), making option A correct.

The governing concept is direct evaluation of an explicit sequence rule at a specified index. The notation a₄ means that n must be replaced by 4 everywhere in the expression, both in the numerator and in the denominator. Thus a₄ = (4² + 1)⁄4 = (16 + 1)⁄4 = 17⁄4. This fraction is already in simplest form because 17 is prime and has no common factor with 4. Therefore option A is correct. Option B could arise from incorrectly subtracting 1 instead of adding it, while option C changes the numerator without any valid operation. Option D equals 18⁄4, so it also does not follow from the given formula. Careful substitution avoids these errors.

The governing concept is the general term of an arithmetic progression, aₙ = a₁ + (n − 1)d. The difference between the thirteenth and sixth terms covers 13 − 6 = 7 equal steps, so 110 − 47 = 7d. Hence 63 = 7d and d = 9. Using the sixth term, 47 = a₁ + 5(9), so a₁ = 2. Therefore aₙ = 2 + (n − 1)9 = 2 + 9n − 9 = 9n − 7, making option B correct. A direct check gives a₆ = 54 − 7 = 47 and a₁₃ = 117 − 7 = 110. Option A has the wrong constant, option C has the wrong common difference, and option D fails both the coefficient and the given terms.

The governing idea is substitution into an explicit general term. The formula aₙ = 11n − 15 is valid for any permitted index, so replace the index n by the entire expression 2r − 1. Thus a₂ᵣ₋₁ = 11(2r − 1) − 15. Distributing 11 gives 22r − 11 − 15, and combining constants gives 22r − 26. Therefore option A is correct. The complete index must be substituted; replacing it only by r would produce the wrong expression. Option B forgets to subtract the contribution from −1, option C uses 11r instead of 22r, and option D adds rather than correctly combines the constant terms.

This is a geometric progression because each term is obtained by multiplying the preceding term by 5: 30/6 = 5, 150/30 = 5, and 750/150 = 5. For a geometric sequence, aₙ = arⁿ⁻¹, where a is the first term and r is the common ratio. Here a = 6 and r = 5, so a₆ = 6 × 5⁵. Since 5⁵ = 3125, a₆ = 6 × 3125 = 18750. Thus option A is correct. The exponent is 5 rather than 6 because the first term already contains the initial factor. Options C and D use an insufficient or incorrect multiplication, while option B does not follow the progression.

The governing pattern is a quadratic sequence because its first differences are 13, 19, 25, and 31, while the second differences are all 6. A quadratic expression is therefore expected. Test option C at successive positive integer values of n: for n = 1, 3(1)² + 4(1) + 1 = 8; for n = 2, 3(2)² + 4(2) + 1 = 21; and for n = 3, 3(3)² + 4(3) + 1 = 40. It also gives 65 when n = 4 and 96 when n = 5. Thus the general term is aₙ = 3n² + 4n + 1, so option C is correct. Each other option fails at least one of the initial terms, so it cannot represent the whole sequence.

Direct answer: option A, 7n − 2. The arithmetic-sequence formula is aₙ = a₁ + (n − 1)d. Since the ninth term is eight steps after the first term, a₉ = a₁ + 8d. Substitute a₉ = 61 and d = 7: 61 = a₁ + 8(7) = a₁ + 56, so a₁ = 5. Therefore aₙ = 5 + (n − 1)7 = 5 + 7n − 7 = 7n − 2. Verification at n = 9 gives 7(9) − 2 = 63 − 2 = 61. Option B gives 68 at n = 9, and option C gives 75, so neither meets the condition. Option D has coefficient 9, but the coefficient of n in an arithmetic nth-term formula must equal the common difference, which is 7. The main indexing warning is that a₉ has eight gaps from a₁, not nine.

The governing concept is direct substitution into an explicit sequence formula followed by subtraction. For n = 8, a₈ = 4(8²) + 7(8) = 4(64) + 56 = 256 + 56 = 312. For n = 3, a₃ = 4(3²) + 7(3) = 4(9) + 21 = 36 + 21 = 57. Therefore a₈ − a₃ = 312 − 57 = 255, so option B is correct. The order matters: the question asks for the eighth term minus the third term, not the reverse. Values such as 245, 265, or 275 may arise from an incorrect square, an inaccurate multiplication, or a subtraction error. The stated formula independently confirms 255.

The governing concept is the constant first difference of a linear sequence. Let the common difference be d. From the fifth term to the twelfth term, the index increases by 12 − 5 = 7, while the value increases by 58 − 23 = 35. Therefore, 7d = 35 and d = 5. From the twelfth term to the twentieth term there are 20 − 12 = 8 steps, so a₂₀ = 58 + 8 × 5 = 98. Equivalently, aₙ = 5n − 2, which also gives a₂₀ = 100 − 2 = 98. Hence option A is correct. The other choices can result from counting seven rather than eight steps or making an arithmetic error.

The governing concept is the general linear rule aₙ = rn + s, where r is the constant change per increase of one in the index. Between n = 3 and n = 10, the term value rises from 14 to 49, an increase of 35 across 7 index intervals. Thus 7r = 35, so r = 5. Substituting a₃ = 14 gives 3(5) + s = 14, hence s = −1. Therefore a₁₅ = 15(5) − 1 = 75 − 1 = 74. The same result follows by moving five steps from a₁₀: 49 + 5 × 5 = 74. Thus option B is correct. The other options reflect an incorrect rate, offset, or number of steps.

The governing concept is construction of an explicit nth-term rule from magnitude and sign patterns. The absolute values are 4, 7, 10, 13, 16, increasing by 3 each time. Their nth-term rule is 3n + 1, since n = 1 gives 4 and each increase of n adds 3. The signs alternate, beginning with positive. The factor (−1)ⁿ⁺¹ is positive at n = 1, negative at n = 2, and continues this pattern. Therefore the general term is (−1)ⁿ⁺¹(3n + 1), so option A is correct. Option B makes the first term negative, C omits the alternating signs, and D has an incorrect magnitude rule.