Explicit or general rule

Use the given general rule \(a_n=11n-6\) separately for the two required positions. For \(n=7\), \(a_7=11(7)-6=77-6=71\). For \(n=2\), \(a_2=11(2)-6=22-6=16\). The requested expression is \(a_7-a_2\), so its value is \(71-16=55\). This also agrees with the fact that moving from the second to the seventh position covers five steps of size 11.

The answer must be the difference, not the value of \(a_7\) alone or the value of \(a_2\) alone. Careful substitution and subtraction lead to 55. Since 55 appears in option B, option B is correct. The other numerical choices do not result from the stated rule and the requested subtraction.

The magnitude is (5n) and signs start positive and alternate, so (a_n=(-1)^{n+1}5n). Choose the power of ((-1)) by checking the first term sign.

(3n^2+4n) does not give the sequence; the correct rule is (4n^2+3). Options should be matched with the first four terms.

For a term to be negative, \(15-3n<0\). This gives \(n>5\), so the smallest integer value is \(n=6\). In fact, \(a_5=15-3(5)=0\), which is not negative, whereas \(a_6=15-3(6)=-3\). Hence, the sixth term is the first negative term. Exam tip: when asked for the first such term, choose the smallest natural-number index satisfying the inequality.

The increase over five gaps is (25), so (d=5), hence (a_{12}=42+3(5)=57). Extend the terms using the common difference.

For option D, substituting \(n=1,2,3,4\) gives \(10,21,34,49\), respectively. Hence, the correct general term is \(a_n=n^2+8n+1\). Option A gives the first term as 10, but for \(n=2\) it gives 20, not 21. Exam tip: When first differences are not constant, use the second differences to test a quadratic form \(an^2+bn+c\).

To find the fifth term, substitute \(n=5\) in the rule: \(a_5=(2\times5-1)^2+2=9^2+2=81+2=83\). Therefore, 83 is the correct option. The value 81 is only \(9^2\); the additional 2 in the rule must also be included. Exam tip: substitute the value of \(n\) first, then simplify brackets, powers, and addition in order.

The change over five gaps is (-15), so (d=-3), hence (a_n=21-3n). From two given terms first find the common difference.

The governing concept is using finite differences to identify the type of explicit rule. The first differences are 19 − 5 = 14, 45 − 19 = 26, and 83 − 45 = 38. The second differences are 26 − 14 = 12 and 38 − 26 = 12, which are constant, indicating a quadratic rule. Now test option A: for n = 1, 6(1)² − 4(1) + 3 = 5; for n = 2, 24 − 8 + 3 = 19; for n = 3, 54 − 12 + 3 = 45; and for n = 4, 96 − 16 + 3 = 83. It matches all terms, so A is correct. Options B and D are linear, while option C gives 26 at n = 2 rather than 19.

The governing concept is testing a proposed explicit rule against all available sequence terms, with finite differences helping identify a quadratic pattern. The first differences are 7, 11, and 15, and the second differences are 4 and 4, so a quadratic expression is reasonable. Test option A: at n = 1, 2(1)^2 + 1 + 2 = 5; at n = 2, 2(2)^2 + 2 + 2 = 12; at n = 3, 2(3)^2 + 3 + 2 = 23; and at n = 4, 2(4)^2 + 4 + 2 = 38. It reproduces every listed term, so A is correct. Option B gives 5, 12, 21, 32; C gives 5, 11, 21, 35; and D gives 5, 11, 17, 23. Each distractor fails on at least one early term.

The governing concept is recognizing and verifying a quadratic explicit rule. The first differences are 13, 23, and 33; their second differences are both 10, which is consistent with a quadratic expression whose n^2 coefficient is 5. Test option A directly: at n = 1, 5 - 2 + 6 = 9; at n = 2, 20 - 4 + 6 = 22; at n = 3, 45 - 6 + 6 = 45; and at n = 4, 80 - 8 + 6 = 78. It reproduces every given term, so option A is correct. Option B gives 9, 20, 41, 72; C gives 9, 24, 47, 78; and D gives 9, 25, 47, 75. Matching only the first term is insufficient, so later substitution distinguishes the correct rule.

Direct answer: option A. A general term must reproduce the listed value at every corresponding position. Test n = 1, 2, 3, and 4 in each promising expression. For A, n = 1 gives 1(4 + 1)/2 = 5/2. At n = 2 it gives 2(8 + 1)/2 = 18/2 = 9. At n = 3 it gives 3(12 + 1)/2 = 39/2. At n = 4 it gives 4(16 + 1)/2 = 68/2 = 34. Therefore A reproduces all four terms. B gives 5/2 when n = 1 but gives 2(6)/2 = 6 when n = 2, not 9. C gives 3 when n = 1, so it fails immediately. D gives (3 + 1)/2 = 2 for the first term, also incorrect. The correct rule must fit the whole sequence, not merely one value. A useful check is to test the first two or three positions before accepting a formula.

An explicit rule must generate each term from n without needing the preceding term. Test option A: at n = 1, it gives 7·2⁰ + 4 = 11; at n = 2, it gives 7·2¹ + 4 = 18; at n = 3, it gives 7·2² + 4 = 32; and at n = 4, it gives 7·2³ + 4 = 60. Therefore A exactly matches the sequence. Option B gives 11, 22, 33, 44 and is not the sequence. Option C begins 11 but gives 13 at n = 2, while option D gives 11 at n = 1 but 23 at n = 2. Matching several positions, rather than only the first term, establishes the answer.

Direct answer: option A. To verify an explicit rule, substitute the position numbers n = 1, 2, 3, and 4. In A, n = 1 gives (3 + 5)/4 = 2. For n = 2, the numerator is 3(4) + 5(2) = 22, so the value is 22/4 = 11/2. For n = 3, the numerator is 3(9) + 5(3) = 42, giving 42/4 = 21/2. For n = 4, the numerator is 3(16) + 5(4) = 68, giving 68/4 = 17. Thus A matches every listed term. B gives 2 when n = 1 but gives 5 when n = 2, not 11/2. C gives 2 at n = 1 but gives 5 at n = 2, so it also fails. D gives (5 + 1)/4 = 3/2 at n = 1, not 2. A formula is accepted only when it reproduces the sequence consistently. Keep the square on n separate from the coefficient while calculating.

Direct answer: option B, 48 : 13. First evaluate the two requested terms separately. For n = 4, a₄ = 6(4²) − 4 + 4 = 6×16 = 96. For n = 2, a₂ = 6(2²) − 2 + 4 = 24 − 2 + 4 = 26. Therefore a₄ : a₂ = 96 : 26. A ratio is normally written in simplest form, and both parts are divisible by 2. Dividing both by 2 gives 48 : 13. Hence B is the required answer. Option A is the correct unsimplified ratio, but it is not in simplest form. Option C reverses the requested order: it represents a₂ : a₄ after simplification, not a₄ : a₂. Option D does not result from correct substitution and arithmetic. The key warning is to calculate each term before simplifying the ratio; do not simplify the formula itself in a way that changes the order or value.

The governing concept is finding an explicit rule for a sequence by examining differences and then verifying the formula. The first differences are 26−9=17, 55−26=29, and 96−55=41. The second differences are 12 and 12, which indicates a quadratic rule. Test option A: when n=1, 6(1)²−1+4=9; when n=2, 6(2)²−2+4=26; when n=3, 6(3)²−3+4=55; and when n=4, 6(4)²−4+4=96. Thus it reproduces every listed term and is the unique correct answer. Option B is linear and cannot have changing first differences. Option C gives 9 at n=1 but fails at n=3, while option D gives 9 at n=1 but gives 28 rather than 26 at n=2.

The governing concept is an explicit rule containing exponential growth. Substitute the position n into option A: for n = 1, 5¹ + 1 − 4 = 2; for n = 2, 5² + 2 − 4 = 23; for n = 3, 5³ + 3 − 4 = 124; and for n = 4, 5⁴ + 4 − 4 = 625. Thus the formula exactly generates the four given terms, so A is correct. Option B is linear and gives 7 for the first term, option C gives 7 for the first term, and option D gives −3 for n = 1 and 28 for n = 2. The small adjustment n − 4 must not be omitted when checking the power pattern.

The governing concept is finding an explicit quadratic rule from the sequence. The first differences are 15, 19 and 23, which increase by 4, so the second difference is constant and a quadratic formula is expected. Check option A: for n = 1, 2 + 9 + 1 = 12; for n = 2, 8 + 18 + 1 = 27; for n = 3, 18 + 27 + 1 = 46; and for n = 4, 32 + 36 + 1 = 69. Therefore A matches all four terms. Option B and C are linear and cannot reflect the changing first differences. Option D gives 11 for n = 1 and 28 for n = 2, so it fails immediately.

The governing concept is the nth-term formula for an arithmetic progression: aₙ = a₁ + (n − 1)d. The first term is a₁ = 7, and the common difference is d = 12 − 7 = 5. Substituting these values gives aₙ = 7 + (n − 1)5 = 7 + 5n − 5 = 5n + 2. Checking n = 1 gives 7, n = 2 gives 12, and n = 3 gives 17, so the expression reproduces the given progression. Hence option B is correct. Option A gives a first term of 7 but a difference of 7; option C gives a first term of 3; and option D gives a first term of 7 but a difference of 2. Only option B satisfies both the initial value and the repeated difference.

Answer: C, aₙ = n² + 4n. The first differences are 7, 9, 11, and 13; they increase by 2, indicating a quadratic rule. Test option C: at n = 1, 1 + 4 = 5; at n = 2, 4 + 8 = 12; at n = 3, 9 + 12 = 21; at n = 4, 16 + 16 = 32; and at n = 5, 25 + 20 = 45. It reproduces every term. Option A gives 3, 8, 15, 24, 35, so it fails at the first term. Option B gives 5, 11, 21, 35, 53, so it fails at the second term. Option D gives 5, 9, 13, 17, 21, so it does not match the sequence. The important distinction is that an explicit nth-term formula must generate terms directly from their positions. Memory cue: calculate differences first, then verify a candidate at n = 1, 2, and 3.

Direct answer: option A, 2ⁿ⁺² − 1. Look for powers of 2 by adding 1 to each term: 7 + 1 = 8 = 2³, 15 + 1 = 16 = 2⁴, 31 + 1 = 32 = 2⁵, 63 + 1 = 64 = 2⁶, and 127 + 1 = 128 = 2⁷. The first term, corresponding to n = 1, uses exponent 3. Therefore the exponent is n + 2, giving aₙ = 2ⁿ⁺² − 1. Checking n = 1 gives 2³ − 1 = 7, and n = 5 gives 2⁷ − 1 = 127. Option B uses exponent n + 1, so it gives 3 for the first term rather than 7. Option C does not keep the doubling pattern: at n = 2 it gives 9, not 15. Option D is linear and cannot describe the rapidly doubling terms. Memory cue: add 1 first; the resulting powers are 2³, 2⁴, 2⁵, so start the exponent at n + 2.