Recursive rule

Correct answer: B. 15

Explanation: In a recursive rule, substitute a new value of n for each next term. For n=1, a_2=9+2(1)-2=9; for n=2, a_3=9+2(2)-2=11; and for n=3, a_4=11+2(3)-2=15. Therefore, the correct answer is 15. The value 17 would result from incorrectly using an increment of 6 instead of 4 in the last step. Exam tip: To find a_4, apply the rule successively for n=1, 2, and 3.

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

Explanation: In option A, each new term is formed from the previous term \(a_{n-1}\), and an initial term \(a_1\) is given, so it is recursive. Rules such as \(a_n=2n+1\) are explicit. Exam tip: look for a previous-term reference.

Correct answer: B. 10

Explanation: Using the recursive rule successively, \(a_2=5+\frac{1}{2}\), \(a_3=a_2+\frac{2}{2}\), \(a_4=a_3+\frac{3}{2}\), and \(a_5=a_4+\frac{4}{2}\). Hence, \(a_5=5+\frac{1+2+3+4}{2}=5+5=10\). Therefore, 10 is correct. Choosing 9 would omit the final increase \(\frac{4}{2}\). Exam tip: to find \(a_5\), add the increments for \(n=1\) through \(n=4\).

Correct answer: B. 27

Explanation: In the recursive rule, use \(n=1,2,3\) successively to reach the fourth term. Thus, \(a_2=30-\frac12\), \(a_3=30-\frac12-1\), and \(a_4=30-\frac12-1-\frac32=30-3=27\). Therefore, 27 is correct. The value 26 would result from adding the subtractions incorrectly. Exam tip: to find \(a_4\) from \(a_1\), apply the rule three times.

Correct answer: B. (16)

Explanation: (a_2=1\times2+1=3), (a_3=2\times3+1=7), and (a_4=3\times7+1=22). The correct option should include (22).

Correct answer: C. 33

Explanation: In a recursive rule, substitute the appropriate value of n at every step. Thus, a_2=5+2(1)+2=9, a_3=9+2(2)+2=15, a_4=15+2(3)+2=23, and a_5=23+2(4)+2=33. Therefore, 33 is correct. A value such as 31 results from using an incorrect increment in one of the steps. Exam tip: to find a_5, apply the rule successively for n=1 through n=4.

Correct answer: B. 39

Explanation: Using n=1 in the recursive rule gives a_2=60-(3×1+1)=56. Then, for n=2, a_3=56-7=49, and for n=3, a_4=49-10=39. Therefore, the correct answer is 39. A value such as 42 can result from calculating the amount to be subtracted incorrectly. Exam tip: to find a_4, apply the rule successively for n=1, 2, and 3.

Correct answer: C. 42

Explanation: The governing concept is step-by-step evaluation of an index-dependent recurrence. Begin with a₁ = 3 and substitute n = 1 to obtain a₂ = 2a₁ + 1 + 1 = 2(3) + 2 = 8. Next use n = 2 and the newly calculated term: a₃ = 2a₂ + 2 + 1 = 2(8) + 3 = 19. Finally, use n = 3 to calculate the fourth term: a₄ = 2a₃ + 3 + 1 = 2(19) + 4 = 38 + 4 = 42. Therefore option C is correct. The index changes at every step, so the added quantity is 2, then 3, then 4. Values such as 39, 41, or 45 arise from an arithmetic substitution or indexing error. In particular, the recurrence must be applied three times because the first term is already supplied.

Correct answer: B. 53

Explanation: The governing concept is an index-dependent recursive rule. To calculate a_2, use n=1 because the recurrence produces a_{n+1} from a_n: a_2=3a_1−2(1)=3(7)−2=19. To calculate a_3, use n=2 and the newly found value a_2: a_3=3a_2−2(2)=3(19)−4=57−4=53. Therefore option B is correct. The index must change from 1 to 2 in the second application. Reusing n=1, forgetting the factor 3, or subtracting the wrong quantity gives distractor values such as those listed. The recurrence is not a constant-difference rule, so ordinary AP shortcuts cannot replace the stated step-by-step calculation.

Correct answer: A. \(a_1=4,\ a_{n+1}=a_n+3\)

Explanation: A recursive definition needs both an initial term and a rule connecting each new term to a previous term. Option A provides \(a_1=4\) and \(a_{n+1}=a_n+3\). Option C lacks an initial value. Exam tip: check for both parts.

Correct answer: D. 37

Explanation: This is a second-order recursive sequence because each term depends on the two immediately preceding terms. The initial values are a₁ = 2 and a₂ = 5. For n = 3, a₃ = a₂ + 2a₁ = 5 + 2(2) = 9. For n = 4, a₄ = a₃ + 2a₂ = 9 + 2(5) = 19. For n = 5, a₅ = a₄ + 2a₃ = 19 + 2(9) = 37. Therefore option D is correct. The coefficient 2 must multiply the term two positions earlier, not the immediately preceding term. Values 23, 27, and 31 can result from stopping before the fifth term or applying that coefficient to the wrong term. Writing every intermediate term prevents such indexing errors.

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

Explanation: In an arithmetic sequence, consecutive terms have a constant difference. Here \(a_n-a_{n-1}=3\), so each new term is obtained by adding 3 to the previous term. Option B multiplies terms and is geometric in nature. Exam tip: for a fixed difference, use “previous term + d”.

Correct answer: D. 34

Explanation: Using the recursive rule, each new term is twice the previous term minus the term before it. Thus, \(a_3=2(10)-4=16\), \(a_4=2(16)-10=22\), \(a_5=28\), and \(a_6=34\). Therefore, the correct answer is 34. Although 32 is a close distractor, it is not a term of this sequence. Exam tip: write the next few terms in order when evaluating a recursive sequence.

Correct answer: C. 32

Explanation: In a recursive rule, each new term is found by substituting the current value of n into the increment and adding it to the previous term. Thus, a_2=6+(1^2+2×1)=9, a_3=9+(2^2+2×2)=17, and a_4=17+(3^2+2×3)=32. Therefore, 32 is correct. The value 30 may result from incorrectly calculating the third increment instead of using 15. Exam tip: to find a_4, apply the rule successively for n=1, 2, and 3.

Correct answer: B. 70

Explanation: Substitute n=1, 2, and 3 in the recursive rule. The quantities subtracted are 2, 6, and 12 respectively. Thus, a_2=90-2=88, a_3=88-6=82, and a_4=82-12=70. Therefore, 70 is correct. The value 72 can result from using an incorrect value for n^2+n in the third step. Exam tip: to find a_4 from a_1, apply the rule three times.

Correct answer: D. 40

Explanation: In a recursive rule, each new term is obtained by adding the required power to the previous term. Thus, a_2=1+3^1=4, a_3=4+3^2=13, and a_4=13+3^3=40. Therefore, the correct answer is 40. The value 39 would result from incorrectly using 26 instead of the final increment 27. Exam tip: to find a_4, apply the rule successively for n=1, 2, and 3.

Correct answer: A. 61

Explanation: Using n=1, 2, and 3 in the recursive rule, subtract 3, 9, and 27 respectively. Thus, \(a_2=100-3=97\), \(a_3=97-9=88\), and \(a_4=88-27=61\). Therefore, the correct answer is 61. The value 64 would result from subtracting 24 in the last step, but the rule requires subtracting \(3^3=27\). Exam tip: Increase n by 1 for each new term before evaluating the power.

Correct answer: C. 12

Explanation: Apply the recursive rule successively for n=1, 2, and 3. We get a_2=8+2(-1)^1+1=7, a_3=7+2(-1)^2+2=11, and a_4=11+2(-1)^3+3=12. Hence, the correct answer is 12. A value such as 10 can result from mishandling the alternating sign in (-1)^n. Exam tip: write the value of n at every step and check the sign separately.

Correct answer: C. 11

Explanation: First, put n=1: \(a_2=8+2(-1)^1+1=8-2+1=7\). Then put n=2: \(a_3=7+2(-1)^2+2=7+2+2=11\). Therefore, the correct answer is 11. Choosing 10 would incorrectly treat \((-1)^2\) as negative. Exam tip: in a recursive rule, substitute consecutive values of n carefully while finding each next term.

Correct answer: B. 35

Explanation: In the recursive rule, add \(n(n+3)\) to the current term each time. \(a_2=3+1(1+3)=7\), \(a_3=7+2(2+3)=17\), and \(a_4=17+3(3+3)=35\). Hence, the correct answer is 35. The option 36 may result from incorrectly taking the final increase as 19 instead of 18. Exam tip: to find \(a_4\), apply the rule successively for \(n=1,2,3\).

Correct answer: B. 17

Explanation: Given \(a_1=3\) and \(a_{n+1}=a_n+n(n+3)\), first put \(n=1\): \(a_2=3+1(1+3)=7\). Next, put \(n=2\): \(a_3=7+2(2+3)=7+10=17\). Therefore, the correct answer is 17. A result such as 15 can arise from incorrectly evaluating the added term. Exam tip: in a recursive rule, substitute the correct value of \(n\) at every step.

Correct answer: A. 88

Explanation: Substitute n=1, 2, and 3 in the recursive rule. The amounts subtracted are 1(1+3)=4, 2(2+3)=10, and 3(3+3)=18. Thus, a_2=120-4=116, a_3=116-10=106, and a_4=106-18=88. Therefore, 88 is correct. An option such as 90 can result from an incorrect subtraction in the final step. Exam tip: to find a_4, apply the rule only for n=1 through n=3.

Correct answer: C. 67

Explanation: In a recursive rule, each new term is found from the preceding term. Thus, \(a_2=2(4)+5=13\), \(a_3=2(13)+5=31\), and \(a_4=2(31)+5=67\). Therefore, 67 is correct. A value such as 65 can result from applying the rule incorrectly to the previous term. Exam tip: To find \(a_4\) from \(a_1\), apply the rule three times in order.

Correct answer: A. \(a_1=5,\ a_{n+1}=a_n+3\)

Explanation: The first term is 5, and each next term is obtained by adding 3: \(5+3=8\) and \(8+3=11\). Hence A is correct. In B, terms are multiplied by 3 instead. Exam tip: generate two terms to check a recursive rule.

Correct answer: B. 18

Explanation: To find the second term, substitute \(n=1\) and \(a_1=5\) in the recursive rule: \(a_2=3a_1+1+2=3\times5+1+2=18\). Therefore, 18 is correct. Option 16 may result from not adding the \(n+2\) part correctly. Exam tip: while finding \(a_{n+1}\), first substitute the correct value of \(n\).