Recursive rule

Correct answer: C. 39

Explanation: A recursive rule finds each new term from earlier terms. Here the rule is \\(a_n=2a_{n-1}+a_{n-2}\\), so twice the immediately preceding term is added to the term before it. The coefficient 2 does not multiply both terms. Starting with a₁ = 2 and a₂ = 7, we can calculate the terms in order.

First, \\(a_3=2a_2+a_1=2(7)+2=16\\). Next, \\(a_4=2a_3+a_2=2(16)+7=39\\). Thus the fourth term is 39, which is option C. A common error is to use only the previous term or to multiply the sum of the two previous terms by 2; neither follows the stated recurrence.

Correct answer: C. 94

Explanation: Because this is a second-order recurrence, calculate each term from the two immediately preceding terms. First, a₃ = 3(10) − 2(4) = 30 − 8 = 22. Next, a₄ = 3(22) − 2(10) = 66 − 20 = 46. Finally, a₅ = 3(46) − 2(22) = 138 − 44 = 94. Therefore option C is correct; applying both coefficients to the wrong terms changes the result.

Correct answer: B. (52)

Explanation: The added values are (4,13,36), so (a_4=56). Exam tip: calculate the power and square separately.

Correct answer: B. 20

Explanation: Apply the recursive rule step by step. \(a_2=3+3^1+1^2=7\), and \(a_3=7+3^2+2^2=7+9+4=20\). Hence, 20 is correct. A value such as 18 can result from omitting \(2^2\), which is incorrect. Exam tip: use the current value of \(n\) in \(a_{n+1}\); for \(a_3\), take \(n=2\).

Correct answer: A. (100)

Explanation: The subtracted values are (3,8,17,32), so (a_5=90). Exam tip: add all subtractions before subtracting.

Correct answer: B. 122

Explanation: Apply the recurrence for n = 1, 2, and 3. The deductions are 2¹ + 1² = 3, 2² + 2² = 8, and 2³ + 3² = 17. Starting from 150 gives a₂ = 150−3 = 147, a₃ = 147−8 = 139, and a₄ = 139−17 = 122. Hence option B is correct. The other values arise from an arithmetic error or from using the wrong index in the power or square.

Correct answer: C. 18

Explanation: Substituting n=1,2,3,4 in the recursive rule gives: a_2=12-4=8, a_3=8+7=15, a_4=15-10=5, and a_5=5+13=18. Hence, the correct answer is 18. The value 15 is a close distractor because it is a_3, not a_5. Exam tip: track the alternating sign from (-1)^n carefully: it is negative for odd n and positive for even n.

Correct answer: B. 5

Explanation: Apply the recursive rule successively for n=1, 2, and 3. We get a_2=12-4=8, a_3=8+7=15, and a_4=15-10=5. Therefore, the correct answer is 5. The value 7 may result from not applying the third-step change correctly. Exam tip: determine the sign of (-1)^n first at every step, then calculate the next term.

Correct answer: C. (44)

Explanation: The added values are (7,14,23), so (a_4=50). Exam tip: calculate the product and then add (2).

Correct answer: B. 27

Explanation: Use the recursive rule successively to find the next terms. For \(n=1\), \(a_2=6+1(1+4)+2=13\). Then, for \(n=2\), \(a_3=13+2(2+4)+2=13+12+2=27\). Therefore, the correct answer is \(27\). The nearby distractor \(25\) can result from incorrectly evaluating \(2(2+4)\) in the second step. Exam tip: substitute the next integer value of \(n\) each time you calculate a new term.

Correct answer: A. 156

Explanation: The governing concept is a recursive rule with an index-dependent decrement. To move from aₙ to aₙ₊₁, calculate the complete quantity n(n + 4) + 2 and subtract it from the current term. For n = 1, the decrement is 1(1 + 4) + 2 = 7, so a₂ = 200 − 7 = 193. For n = 2, the decrement is 2(2 + 4) + 2 = 14, giving a₃ = 193 − 14 = 179. For n = 3, it is 3(3 + 4) + 2 = 23, so a₄ = 179 − 23 = 156. Hence option A is correct. Omitting the final 2, using the wrong value of n, or subtracting only the product would produce different distractor values.

Correct answer: B. 69

Explanation: Apply the recursive rule successively with the appropriate value of n. We get a_2=2(4)+4(1)-1=11, a_3=2(11)+4(2)-1=29, and a_4=2(29)+4(3)-1=69. Therefore, the correct answer is 69. A value such as 71 can result from using an incorrect value of n in the term 4n-1. Exam tip: to find a_4, apply the rule in order for n=1, 2, and 3.

Correct answer: A. \(a_1=2\) and \(a_{n+1}=a_n+3\), where \(n\geq1\)

Explanation: Option A gives both an initial term, \(a_1=2\), and a rule for obtaining each next term, so it determines one unique sequence. For example, \(a_2=2+3=5\). Exam tip: a recursive definition needs a starting value.

Correct answer: C. 31

Explanation: Apply the recursive rule first with \(n=1\): \(a_2=\frac{96}{2}+1^2+1=48+2=50\). Next, use \(n=2\): \(a_3=\frac{50}{2}+2^2+2=25+6=31\). Hence, 31 is correct. Option 30 may result from adding \(2^2+2\) incorrectly in the second step. Exam tip: To find \(a_3\), apply the rule successively for \(n=1\) and \(n=2\).

Correct answer: A. 7

Explanation: Use the recursive rule step by step. For \(n=1\), \(a_2=\frac{24}{3}+1^2=8+1=9\). Then, for \(n=2\), \(a_3=\frac{9}{3}+2^2=3+4=7\). Therefore, the correct answer is 7. Option 9 is the value of \(a_2\), not \(a_3\). Exam tip: substitute successive values of \(n\) in order when applying a recursive rule.

Correct answer: B. 55

Explanation: Use the recursive rule step by step. First, a_2=10+2(10)+1^2=31. Then, with n=2, a_3=31+2(10)+2^2=31+20+4=55. Hence, the correct answer is 55. The value 56 may result from incorrectly evaluating 2^2 as 5. Exam tip: substitute the correct value of n at each step, while a_1=10 remains fixed.

Correct answer: C. 67

Explanation: The recursive rule requires adding the previous two terms and 3n for the current index. Thus, a_3=6+2+3(3)=17, a_4=17+6+3(4)=35, and a_5=35+17+3(5)=67. Therefore, 67 is correct. An answer such as 62 can result from an error while adding 3(5)=15. Exam tip: find a_3 and a_4 in order before calculating a_5.

Correct answer: D. 35

Explanation: Using the recursive rule with n=3, \(a_3=a_2+a_1+3(3)=6+2+9=17\). Then, for n=4, \(a_4=a_3+a_2+3(4)=17+6+12=35\). Therefore, the correct answer is 35. A value such as 33 can result from calculating the \(3n\) term incorrectly. Exam tip: always substitute the current value of n when finding each new term.

Correct answer: B. 27

Explanation: Apply the recursive rule one step at a time. First, \(a_2=3^2-2(3)+3=9-6+3=6\). Then \(a_3=6^2-2(6)+3=36-12+3=27\). Therefore, the correct answer is 27. The value 36 is only \(6^2\); the remaining terms \(-2a_n+3\) in the rule must also be included. Exam tip: substitute the complete previous term into the rule before finding the next term.

Correct answer: B. 13

Explanation: To find the next term, put n=1 in the recursive rule: a_2=a_1^2-a_1+1. Therefore, a_2=4^2-4+1=16-4+1=13. Option 15 results from an incorrect calculation rather than from the given rule. Exam tip: for a_2, substitute a_1 directly for a_n.

Correct answer: C. (10)th

Explanation: (74-11=63), and (63\div7=9) steps, so the term number is (10). Exam tip: add (1) to the number of steps.

Correct answer: C. (97)

Explanation: For (a_{12}), (8) is added (11) times, so (9+88=97). Exam tip: the rule is applied one less time than the term number.

Correct answer: B. 66

Explanation: Given \(a_1=4\), we get \(a_2=2(4)+7=15\), \(a_3=2(15)+7=37\), and \(a_4=2(37)+7=81\). Therefore, \(a_4-a_2=81-15=66\). Hence, 66 is the correct answer. A value such as 62 results from an error while generating the terms or subtracting them. Exam tip: In a recursive sequence, calculate the required terms in order before finding their difference.

Correct answer: C. 22

Explanation: Given a_1=4, the recursive rule gives a_2=2(4)+7=15 and a_3=2(15)+7=37. Therefore, a_3-a_2=37-15=22. The value 20 would result from an error while calculating the terms or subtracting them. Exam tip: in recursive-sequence questions, calculate the required terms in order before finding their difference.

Correct answer: A. 56

Explanation: The recursive rule says that each new term is obtained by adding the previous two terms and 6. Thus, a_3=4+8+6=18, a_4=8+18+6=32, and a_5=18+32+6=56. Therefore, 56 is correct. A value such as 54 results from an error in using the previous terms or adding 6. Exam tip: write each intermediate term in order when solving a recursive sequence.