Recursive rule

Correct answer: B. (42)

Explanation: The terms are (24,32,\frac{128}{3}), so (a_3=\frac{128}{3}). The correct option should include the fractional value.

Correct answer: C. 24

Explanation: Given \(a_1=18\), put \(n=1\) in the recursive rule: \(a_2=a_1+\frac{a_1}{3}=18+\frac{18}{3}=18+6=24\). Hence, 24 is correct. The value 22 would result from incorrectly taking \(18/3\) as 4, whereas it is 6. Exam tip: to find \(a_{n+1}\), substitute the preceding term directly into the rule.

Correct answer: B. 81

Explanation: Apply the recursive rule step by step: \(a_3=4\times3-3\times1=9\), \(a_4=4\times9-3\times3=27\), and \(a_5=4\times27-3\times9=81\). Hence, 81 is correct. The close distractor 27 is \(a_4\), not \(a_5\). Exam tip: while using a recurrence, keep track of the two previous terms in the correct order.

Correct answer: C. 27

Explanation: This is a second-order recursive sequence because each new term depends on the two preceding terms, not just on one term. The supplied starting values are a₁ = 1 and a₂ = 3. For n = 3, calculate a₃ = 4a₂ − 3a₁ = 4(3) − 3(1) = 12 − 3 = 9. Then substitute a₃ and a₂ when n = 4: a₄ = 4a₃ − 3a₂ = 4(9) − 3(3) = 36 − 9 = 27. Hence option C is correct, and the sequence begins 1, 3, 9, 27. A common mistake is to use a₁ instead of a₂ in the second calculation, reverse the coefficients, or ignore the second previous term. Such errors can produce the other choices, but they do not follow the stated recurrence.

Correct answer: B. (47)

Explanation: (a_2=1\times3+2=5), (a_3=2\times5+2=12), and (a_4=3\times12+2=38). The correct option should include (38).

Correct answer: C. 10

Explanation: The governing concept is recursive evaluation: each term is obtained from the preceding term, and the index n must be substituted separately at every transition. Start with a₁ = 6. For n = 1, (-1)¹(2×1) = -2, so a₂ = 6 - 2 = 4. For n = 2, (-1)²(2×2) = 4, so a₃ = 4 + 4 = 8. For n = 3, the change is (-1)³(6) = -6, giving a₄ = 8 - 6 = 2. For n = 4, the change is (+1)(8) = 8, so a₅ = 2 + 8 = 10. Therefore option C is correct. The other choices can result from ignoring the alternating sign, using the wrong index, or stopping before the fourth update.

Correct answer: D. 40

Explanation: Answer: option D, 40. Because the rule is recursive, calculate each term in order. Begin with a₁ = 4. To find a₂, use n = 1 in the increment 7n - 2: 7(1)-2 = 5, so a₂ = 4+5 = 9. To find a₃, use n = 2: 7(2)-2 = 12, so a₃ = 9+12 = 21. To find a₄, use n = 3: 7(3)-2 = 19, so a₄ = 21+19 = 40. Therefore option D is correct. There are three transitions from a₁ to a₄, so n values 1, 2, and 3 are used. Option A does not include the correct total increase. Options B and C result from arithmetic or indexing mistakes. Do not use n = 4 to calculate a₄ directly; n = 4 would be used in the step that produces a₅ from a₄. Memory cue: in aₙ₊₁ = aₙ + f(n), use the current term's index n.

Correct answer: A. 108

Explanation: Answer: option A, 108. The recurrence says that the amount 5n+4 must be subtracted at each step. Start with a₁ = 150. For a₂, use n=1: subtract 5(1)+4 = 9, giving a₂ = 150-9 = 141. For a₃, use n=2: subtract 5(2)+4 = 14, giving a₃ = 141-14 = 127. For a₄, use n=3: subtract 5(3)+4 = 19, giving a₄ = 127-19 = 108. Equivalently, the total subtraction is 9+14+19 = 42, and 150-42 = 108. Option A is correct. Options B, C, and D do not result from all three required subtractions. They may arise from omitting a step, using the wrong value of n, or adding rather than subtracting. Notice that n=3 is used to move from a₃ to a₄; n=4 would produce a₅. Memory cue: write the index beside every transition before calculating.

Correct answer: B. Fifth term

Explanation: The governing concept is a recursive sequence: each new term is calculated from the preceding term, and the value of n identifies the term currently being used. Start with a₁ = 5. For n = 1, a₂ = a₁ + 1² = 5 + 1 = 6. For n = 2, a₃ = 6 + 2² = 10. For n = 3, a₄ = 10 + 3² = 19. For n = 4, a₅ = 19 + 4² = 35. Therefore, 35 is the fifth term, so option B is correct. The common error is to square the index of the new term rather than the n appearing in the recurrence; that changes the sequence and may lead to another option.

Correct answer: A. a₁ = 20, aₙ₊₁ = aₙ − (3n + 2)

Explanation: Answer: A. A recursive rule gives the first term and explains how to obtain each next term from the preceding term. In option A, a1 = 20. For n = 1, subtract 3(1) + 2 = 5, giving a2 = 20 − 5 = 15. For n = 2, subtract 3(2) + 2 = 8, giving a3 = 15 − 8 = 7. For n = 3, subtract 3(3) + 2 = 11, giving a4 = 7 − 11 = −4. Thus it reproduces every displayed term. Option B subtracts 5 at the first step but 7 at the second, giving 8 rather than 7. Option C starts with 15, so it cannot produce the stated first term. Option D gives a2 = 2(20) − 25 = 15, but then a3 = 2(15) − 25 = 5, not 7. Memory cue: check both the starting value and every transition, not just the first two terms.

Correct answer: D. a₁ = 6, aₙ₊₁ = aₙ + 3n + 2

Explanation: A recursive rule must reproduce the first term and every successive difference. The sequence differences are 11 − 6 = 5, 19 − 11 = 8, and 30 − 19 = 11. These increments follow 3n + 2 for n = 1, 2, and 3. Starting with a₁ = 6 gives the required sequence, so option D is correct. Options A and C give different increments, while B begins with the wrong first term.

Correct answer: B. 131

Explanation: The governing concept is a recursive sequence: each term is calculated from the immediately preceding term, and n is substituted for the position of the term being found. Starting with b₁ = 6, calculate b₂ = 3(6) − 2 = 16. Then b₃ = 3(16) − 3 = 45. Finally, b₄ = 3(45) − 4 = 135 − 4 = 131. Thus option B is correct. The nearby alternatives can result from an arithmetic slip, such as forgetting to subtract the current index, using the wrong preceding term, or applying the subtraction before multiplication incorrectly. The recursive rule must be followed in order.

Correct answer: D. 1

Explanation: This is a recursive sequence with an alternating-sign increment. At each step, evaluate (−1)ⁿ first, then multiply by n. Since s₁ = 3, for n = 2 we have s₂ = 3 + (+1)·2 = 5. For n = 3, s₃ = 5 + (−1)·3 = 2. For n = 4, s₄ = 2 + (+1)·4 = 6. Finally, for n = 5, s₅ = 6 + (−1)·5 = 1. Therefore option D is correct. The important point is that even indices contribute positive amounts and odd indices contribute negative amounts here. Confusing the sign pattern, beginning the recurrence at n = 1, or treating every increment as positive can produce the distractor values.

Correct answer: B. −4

Explanation: Direct answer: F₅ = −4, so option B is correct. Rewrite the rule as Fₙ = Fₙ₋₁ − (2n − 1). This shows that at each stage we subtract an odd number. Starting with F₁ = 20, use n = 2 to find F₂: F₂ = 20 − 4 + 1 = 17. Then F₃ = 17 − 6 + 1 = 12, F₄ = 12 − 8 + 1 = 5, and F₅ = 5 − 10 + 1 = −4. Equivalently, the decreases are 3, 5, 7, and 9, whose sum is 24; therefore 20 − 24 = −4. Option A, −10, subtracts too much or mishandles the signs. Option B, −4, agrees with every transition. Option C, 4, ignores that the final total decrease exceeds 20 by 4. Option D, 8, may result from using the wrong starting index or adding a decrease. Memory cue: rewrite −2n + 1 as −(2n − 1), then the odd-number pattern becomes clear.

Correct answer: B. (3, −6, 12, −24)

Explanation: The governing concept is the recursive rule of a geometric progression: each new term equals the preceding term multiplied by the common ratio. Start with a₁ = 3. Then a₂ = 3 × (−2) = −6, a₃ = −6 × (−2) = 12, and a₄ = 12 × (−2) = −24. Thus the first four terms are (3, −6, 12, −24), making option B correct. The negative ratio explains the alternating signs, while the magnitude doubles at each step. Option A incorrectly uses a positive ratio of 2. Option C treats the ratio as though it were the second term, and option D changes the given first term from 3 to −3. The recursive calculation confirms every entry.