Recursive rule

Correct answer: C. 43

Explanation: The governing idea is successive use of the recurrence with the correct current index at every step. Start with a₁ = 4. For n = 1, the rule gives a₂ = 2a₁ + 1 = 2(4) + 1 = 9. For n = 2, use a₂ to obtain a₃ = 2a₂ + 2 = 2(9) + 2 = 20. For n = 3, calculate a₄ = 2a₃ + 3 = 2(20) + 3 = 43. Therefore option C is correct. The added values are 1, 2, and 3 because those are the indices of the transitions producing a₂, a₃, and a₄. Adding 4 while calculating a₄ applies the rule one step too late. Forgetting the factor 2 or using an incorrect preceding term can explain the other plausible options.

Correct answer: A. 18

Explanation: Apply the recursive rule successively for n=1,2,3,4: a_2=50-5=45, a_3=45-7=38, a_4=38-9=29, and a_5=29-11=18. Therefore, 18 is correct. A value such as 20 can result from using an incorrect value of n or subtracting the wrong amount. Exam tip: to reach a_5 from a_1, apply the rule four times.

Correct answer: A. \(a_{n+1}=a_n+3\)

Explanation: In an arithmetic progression, the difference \(a_{n+1}-a_n\) is constant. Rule A gives this difference as \(3\), so its common difference is 3. In option C, the difference is \(3n\), which changes with \(n\). Exam tip: a fixed number added at each step is the common difference.

Correct answer: D. (50)

Explanation: The added values are (3,8,15,24), and after adding them (a_5=50). Exam tip: write values of (n^2+2n) separately.

Correct answer: B. 80

Explanation: The governing concept is a recursive subtraction with a changing decrement. To move from a₁ to a₅, apply the rule for n = 1, 2, 3, and 4. The amounts subtracted are n² + n: for n = 1, the amount is 1² + 1 = 2; for n = 2, it is 2² + 2 = 6; for n = 3, it is 3² + 3 = 12; and for n = 4, it is 4² + 4 = 20. Step by step, a₂ = 120 − 2 = 118, a₃ = 118 − 6 = 112, a₄ = 112 − 12 = 100, and a₅ = 100 − 20 = 80. Equivalently, the total deduction is 2 + 6 + 12 + 20 = 40, so 120 − 40 = 80. Thus option B is correct. The decrement is not constant, and subtracting only the last amount or using five transitions would be incorrect.

Correct answer: B. 54

Explanation: This is a second-order recurrence: each new term equals the immediately previous term plus twice the term two positions earlier. Start with a₁ = 1 and a₂ = 4. For n = 3, a₃ = a₂ + 2a₁ = 4 + 2(1) = 6. For n = 4, a₄ = a₃ + 2a₂ = 6 + 2(4) = 14. For n = 5, a₅ = a₄ + 2a₃ = 14 + 2(6) = 26. Finally, for n = 6, a₆ = a₅ + 2a₄ = 26 + 2(14) = 54. Therefore option B is correct, and the sequence is 1, 4, 6, 14, 26, 54. The coefficient 2 must multiply the term two positions back, not the newest term. Confusing these positions or using only one previous term can produce the other choices but violates the stated recurrence.

Correct answer: D. 23

Explanation: The governing concept is a second-order recursive rule: each new term is calculated from the two immediately preceding terms. Start with a₁ = 5 and a₂ = 8. Then a₃ = 2(8) − 5 = 11, a₄ = 2(11) − 8 = 14, a₅ = 2(14) − 11 = 17, and a₆ = 2(17) − 14 = 20. Finally, a₇ = 2(20) − 17 = 23. Thus option D is correct. The order of the terms matters: the most recent term is doubled, and the term before it is subtracted. The resulting sequence is 5, 8, 11, 14, 17, 20, 23, which also has a constant increase of 3 and independently confirms the result. The other options are earlier terms or arise from an arithmetic or indexing mistake.

Correct answer: C. (24)

Explanation: The added values are (3,6,11), so (a_4=2+3+6+11=22). Exam tip: calculate both (2^n) and (n) before adding.

Correct answer: D. 23

Explanation: The governing concept is repeated use of an index-dependent recurrence. Four transitions are needed to move from a₁ to a₅, so evaluate 2ⁿ − n for n = 1, 2, 3, and 4. The increments are 2¹ − 1 = 1, 2² − 2 = 2, 2³ − 3 = 5, and 2⁴ − 4 = 12. Starting from a₁ = 3 gives a₂ = 3 + 1 = 4, a₃ = 4 + 2 = 6, a₄ = 6 + 5 = 11, and a₅ = 11 + 12 = 23. Equivalently, a₅ = 3 + 1 + 2 + 5 + 12 = 23. Therefore option D is correct. Option C may result from an addition error, while applying the rule five times would incorrectly introduce a transition beyond the requested fifth term.

Correct answer: A. 2

Explanation: Substitute n=1, 2, and 3 successively in the recursive rule. a_2=6+(-1)^1(2)=4, a_3=4+(-1)^2(4)=8, and a_4=8+(-1)^3(6)=2. Therefore, the correct answer is 2. Although 4 is a close distractor, it is only the second term. Exam tip: write the value of n at every step, since (-1)^n is negative for odd n and positive for even n.

Correct answer: B. 30

Explanation: Apply the recursive rule successively for n=1, 2, and 3: a_2=4+1(2)+2=8, a_3=8+2(3)+2=16, and a_4=16+3(4)+2=30. Therefore, the correct answer is 30. The value 28 results if the final +2 is omitted. Exam tip: to find a_4 from a_1, apply the rule three times.

Correct answer: A. (64)

Explanation: A recursive rule gives each new term from the term immediately before it. To find the fourth term, begin with the first term and apply the rule for n = 1, then n = 2, and finally n = 3. The subtraction changes at every step because the value of n changes.

For n = 1, subtracting gives 1(2) + 2 = 4, so a₂ = 90 − 4 = 86. For n = 2, subtract 2(3) + 2 = 8, giving a₃ = 86 − 8 = 78. For n = 3, subtract 3(4) + 2 = 14, giving a₄ = 78 − 14 = 64. Equivalently, the total subtraction is 4 + 8 + 14 = 26, so a₄ = 90 − 26 = 64. Therefore option A is correct. The explanation supplied is accurate, although showing the intermediate terms makes the method clearer.

Correct answer: D. 53

Explanation: In a recursive rule, each next term is found using the previous term and its index. \(a_2=2(3)+2(1)+1=9\), \(a_3=2(9)+2(2)+1=23\), and \(a_4=2(23)+2(3)+1=53\). Therefore, the correct answer is 53. A value such as 49 can result from using an incorrect value of \(n\) in the term \(2n+1\). Exam tip: write the current value of \(n\) at every step while applying a recursive formula.

Correct answer: C. The differences between consecutive terms increase by 2, so it is not an AP

Explanation: From the rule, \(a_{n+1}-a_n=2n\). The differences are 2, 4, 6, ... and are not constant, so the sequence is not an AP. Exam tip: write consecutive differences first to test for an AP.

Correct answer: C. 32.5

Explanation: Apply the recurrence successively, using n = 1 to calculate a₂ and n = 2 to calculate a₃. First, a₂ = a₁/2 + 3(1) = 100/2 + 3 = 53. Next, a₃ = a₂/2 + 3(2) = 53/2 + 6 = 26.5 + 6 = 32.5. Therefore option C is correct. The nearby integer choices result from mishandling the fraction or the index.

Correct answer: B. 13

Explanation: To find \(a_2\), substitute \(n=1\) and \(a_1=20\) into the recursive rule: \(a_2=\frac{20}{2}+3(1)=10+3=13\). Hence, 13 is correct. The value 11 would result from incorrectly adding 1 instead of \(3n\). Exam tip: when finding the next term from \(a_{n+1}\), substitute the correct value of \(n\) first.

Correct answer: C. 30

Explanation: Using the recursive rule with n=1, a_2=5+2(5)+1^2=16. Then, with n=2, a_3=16+2(5)+2^2=16+10+4=30. Therefore, the correct answer is 30. Option 29 is a close distractor, but it is 1 less than the value obtained when 2^2=4 is added correctly. Exam tip: For each next term, substitute the current value of n and keep a_1 fixed as 5.

Correct answer: A. A sequence of perfect squares

Explanation: The added term \(2n+1\) is the difference between consecutive squares: \((n+1)^2-n^2=2n+1\). Hence the terms are \(1,4,9,16,\dots\), or perfect squares. The difference is not constant, so it is not an AP. Exam tip: remember that square differences are consecutive odd numbers.

Correct answer: C. 22

Explanation: Using the recursive rule, first find the third term: \(a_3=a_2+a_1+2(3)=3+2+6=11\). Then \(a_4=a_3+a_2+2(4)=11+3+8=22\). Therefore, the correct answer is 22. A value such as 21 can result from an error while evaluating the \(2n\) term. Exam tip: write the current value of \(n\) explicitly before substituting it into a recursive rule.

Correct answer: A. \(a_n=a_{n-1}+a_{n-2}\)

Explanation: In \(a_n=a_{n-1}+a_{n-2}\), each new term depends on two preceding terms, so both \(a_1\) and \(a_2\) are needed. The other rules use only \(a_{n-1}\), so one initial term is enough. Exam tip: count the earlier terms used in the rule.

Correct answer: B. 4

Explanation: Apply the recursive rule first to \(a_1=4\): \(a_2=4^2-3(4)=16-12=4\). Using \(a_2=4\) again, \(a_3=4^2-3(4)=4\). Hence, the correct answer is 4. The option 8 is incorrect because each term is obtained by subtracting three times the previous term from its square. Exam tip: To find \(a_3\), calculate \(a_2\) first.

Correct answer: B. 58

Explanation: Use the recursive rule step by step, finding each term from the preceding term. First, \(a_2=3^2-3+2=8\). Then \(a_3=8^2-8+2=64-8+2=58\). Therefore, the correct answer is 58. The value 64 is only \(8^2\); it ignores the \(-a_n+2\) part of the rule. Exam tip: To find \(a_3\), calculate \(a_2\) first rather than substituting \(a_1\) directly.

Correct answer: B. 8

Explanation: Put \(n=1\) in the recursive rule: \(a_2=a_1^2-a_1+2\). Hence, \(a_2=3^2-3+2=9-3+2=8\). Therefore, 8 is correct. Option 6 can result from an error in calculating \(3^2\) or in the final addition. Exam tip: to find \(a_2\), always substitute \(n=1\) in the recurrence relation.

Correct answer: C. (52)

Explanation: For (a_{10}), (5) is added (9) times, so (7+45=52). Exam tip: the rule is applied one less time than the term number.

Correct answer: A. \(a_1=5,\ a_{n+1}=2a_n\)

Explanation: In a geometric sequence, each term is obtained by multiplying the preceding term by a fixed ratio. Here the ratio is 2, so \(a_{n+1}=2a_n\). Check: the second term is \(2\times5=10\). In exams, “twice” means multiply by 2, not add 2.