Recursive rule

Correct answer: C. 32

Explanation: Apply the recursive rule using the preceding term each time: \(N_2=2(8)-2(1)=14\), \(N_3=2(14)-3(2)=22\), and \(N_4=2(22)-4(3)=32\). Hence, the correct answer is \(32\). A close distractor such as \(34\) can result from evaluating \(4(3)\) incorrectly. Exam tip: calculate \(n(n-1)\) separately at every step before subtracting it.

Correct answer: B. (34)

Explanation: The direct answer is B, 34. The rule is r_n=2r_{n-1}-r_{n-2}+n, so each new term uses twice the immediately previous term, subtracts the term before that, and then adds the current index. Given r_1=4 and r_2=6: r_3=2(6)-4+3=11. Then r_4=2(11)-6+4=20. Finally r_5=2(20)-11+5=34. Option A, 30, omits or mishandles part of the calculation. Option B is correct because it follows the exact order of operations and subtracts r_3 when finding r_5. Option C, 38, and option D, 42, are inconsistent with the recurrence. The important point is that r_{n-2} is subtracted, not r_{n-1}; also, the current n is added. Memory cue: double the newer term, subtract the older term, add the index.

Correct answer: D. 9

Explanation: The governing concept is recursive evaluation with alternating additions and subtractions of consecutive odd numbers. Begin with s₁ = 5. For n = 2, (−1)³ = −1 and 2n − 1 = 3, so s₂ = 5 − 3 = 2. For n = 3, the sign is positive and the amount is 5, giving s₃ = 7. For n = 4, subtract 7 to obtain s₄ = 0. For n = 5, add 9, so s₅ = 9. Therefore option D is correct. The sequence of changes is −3, +5, −7, +9; confusing the exponent n + 1 or treating every change as positive leads to the other options. Writing the sign and magnitude separately prevents that error.

Correct answer: A. 5

Explanation: Using the recursive rule, t_2=2x+3, t_3=2(2x+3)+3=4x+9, and t_4=2(4x+9)+3=8x+21. Since t_4=61, we get 8x+21=61, so 8x=40 and x=5. If x were 6, then t_4 would be 69, not 61. Exam tip: Expand the recurrence step by step up to the given term before solving.

Correct answer: C. Rule (v_n=2v_{n-1}+3)

Explanation: Each new term is obtained by doubling the previous term and adding (3). In exams, check the rule, not only the difference.

Correct answer: B. 102

Explanation: In the recursive rule, each new term is obtained by adding the cube of the current index to the previous term. Thus, (v_2=3+2^3=11), (v_3=11+3^3=38), and (v_4=38+4^3=38+64=102). Therefore, 102 is correct. An answer such as 108 may result from an error while adding the cubes or using the previous term. Exam tip: Start from n=2 after the initial term and calculate each term in order.

Correct answer: D. 43

Explanation: The recursive rule adds \(3^{n-1}+1\) to the preceding term at each step. Thus, \(w_2=1+3^1+1=5\), \(w_3=5+3^2+1=15\), and \(w_4=15+3^3+1=43\). Therefore, the correct answer is 43. A value such as 41 may result from forgetting to add \(+1\) in the final step. Exam tip: substitute each value of \(n\) separately and calculate the terms in order.

Correct answer: A. 64

Explanation: Apply the recursive rule successively for n=2, 3, and 4. x_2=2(4)+2^2-2=10, x_3=2(10)+3^2-3=26, and x_4=2(26)+4^2-4=64. Hence, the correct answer is 64. A value such as 68 can result from evaluating 4^2-4 incorrectly in the final step. Exam tip: substitute the value of n first and simplify the added term separately at every step.

Correct answer: C. पचासी

Explanation: Given \(y_1=2\), we get \(y_2=2+(2\times2-1)^2=2+9=11\), \(y_3=11+(2\times3-1)^2=11+25=36\), and \(y_4=36+(2\times4-1)^2=36+49=85\). Hence, the correct answer is eighty-five. Eighty-one can result from not applying the square correctly in the final step. Exam tip: in a recursive sequence, substitute each value of \(n\) step by step.

Correct answer: B. 137

Explanation: Under the recursive rule, each new term is found by multiplying the previous two terms and adding its term number. Thus, \(z_3=(3\times1)+3=6\), \(z_4=(6\times3)+4=22\), and \(z_5=(22\times6)+5=137\). Therefore, the correct answer is \(137\). A value such as \(145\) can result from using an incorrect previous term or term number. Exam tip: write the two preceding terms at every step before multiplying and adding the current \(n\).

Correct answer: D. 34

Explanation: Using the recursive rule, first put \(n=2\): \(A_2=(2+1)\times 2+2=8\). Then put \(n=3\): \(A_3=(3+1)\times 8+2=34\). Therefore, the correct answer is 34. The option 32 may result from forgetting the final \(+2\). Exam tip: write the value of \(n\) separately at each step before applying the multiplier.

Correct answer: A. 30

Explanation: Apply the recursive rule step by step: \(B_2=11+(2^2-2)=13\), \(B_3=13+(2^3-3)=18\), and \(B_4=18+(2^4-4)=30\). Hence, the correct answer is 30. A value such as 32 can result from calculating \(2^n-n\) incorrectly at one of the steps. Exam tip: find the terms successively for \(n=2,3,4\) rather than jumping directly from \(B_1\) to \(B_4\).

Correct answer: C. 51

Explanation: The recursive rule gives each new term by adding n(n+2) to the preceding term. Thus, C_2=4+2(4)=12, C_3=12+3(5)=27, and C_4=27+4(6)=51. Therefore, the correct answer is 51. A value such as 54 can result from using an incorrect value of n for a step. Exam tip: To find C_4, apply the rule successively for n=2, 3, and 4.

Correct answer: B. 91

Explanation: Use the index of the term being calculated in the recursive rule. Thus, \(D_2=3(5)-(2+1)=12\), \(D_3=3(12)-(3+1)=32\), and \(D_4=3(32)-(4+1)=96-5=91\). Therefore, the correct answer is 91. A value such as 95 can result from incorrectly subtracting 1 instead of \(n+1\). Exam tip: write the value of \(n\) at each step before evaluating \(n+1\).

Correct answer: D. 31

Explanation: In the recursive rule, each new term is obtained by adding the previous two terms and the current index n. Thus, E_3=2+4+3=9, E_4=4+9+4=17, and E_5=9+17+5=31. Therefore, the correct answer is 31. A value such as 29 may result from omitting +5 in the last step. Exam tip: use the subscript of the term as the value of n at each step.

Correct answer: A. 52

Explanation: The recursive rule requires subtracting the value based on the current index at every step. Thus, \(F_2=90-2(3)=84\), \(F_3=84-3(4)=72\), and \(F_4=72-4(5)=52\). Therefore, the correct answer is 52. A value such as 56 may result from subtracting an incorrect value instead of \(4(5)\). Exam tip: to find \(F_4\), substitute \(n=2,3,4\) in order and write each intermediate term.

Correct answer: C. 51

Explanation: In a recursive rule, substitute the index of the term currently being found. Thus, \(G_2=2(1)+2(2)+1=7\), \(G_3=2(7)+2(3)+1=21\), and \(G_4=2(21)+2(4)+1=51\). Therefore, the correct answer is \(51\). A value such as 47 can result from adding the \(2n+1\) term incorrectly. Exam tip: at each step, first double the previous term, then substitute the current value of \(n\) and add \(2n+1\).

Correct answer: B. \(a_n=3a_{n-1}-a_{n-2}\)

Explanation: Option B uses both \(a_{n-1}\) and \(a_{n-2}\), so two starting values, \(a_1\) and \(a_2\), are needed. A and D are first-order rules. Exam tip: check the farthest previous subscript used.

Correct answer: D. (4)

Explanation: The direct answer is D, 4. The recurrence is J_n=J_{n-1}-4(n-1)+1, with J_1=40. At every step, take the previous term, subtract four times one less than the current index, and then add 1. For n=2: J_2=40-4(1)+1=37. For n=3: J_3=37-4(2)+1=30. For n=4: J_4=30-4(3)+1=19. For n=5: J_5=19-4(4)+1=4. Option A, -2, subtracts too much or misses the final addition. Option B, 0, is not the result of the n=5 step. Option C, 2, is close but still misses the exact arithmetic. Option D is correct. Exam cue: substitute the new n first, then calculate the subtraction and the final +1.

Correct answer: A. (89)

Explanation: (K_3=13), (K_4=34), and (K_5=89). In exams, triple the previous term and subtract the term before it.

Correct answer: C. 58

Explanation: The recursive rule forms each new term by adding (5n-4) to the preceding term. Thus, L_2=4+6=10, L_3=10+11=21, L_4=21+16=37, and L_5=37+21=58. Therefore, 58 is correct. The option 55 can result from using an incorrect final increment instead of 21. Exam tip: To find L_5, apply the increments successively for n=2 through n=5.

Correct answer: B. 21

Explanation: In a recursive rule, first find the next term from the previous one. \(M_2=4M_1-2=4\times2-2=6\). Then \(M_3=4M_2-3=4\times6-3=21\). Therefore, the correct answer is 21. The value 25 would result from forgetting to subtract 3 in the final step. Exam tip: use the current index \(n\) each time you apply the rule.

Correct answer: D. Eighty-three

Explanation: In a recursive rule, use the index of the term being calculated at every step. \(R_2=(2+2)\times1-1=3\), \(R_3=(3+2)\times3-1=14\), and \(R_4=(4+2)\times14-1=83\). Therefore, the correct answer is eighty-three. A distractor such as seventy-nine may result from not changing the factor \(n+2\) correctly at each step. Exam tip: calculate \(R_2\) and \(R_3\) in order before finding \(R_4\).

Correct answer: A. 51

Explanation: Apply the recursive rule step by step: \(O_2=5+2^2+2(2)=13\), \(O_3=13+2^3+2(3)=27\), and \(O_4=27+2^4+2(4)=51\). Therefore, the correct answer is 51. A value such as 55 can result from an error while evaluating the \(2n\) term. Exam tip: calculate \(2^n\) and \(2n\) separately at each step before adding them.

Correct answer: C. 84

Explanation: Apply the recursive rule successively: \(P_2=P_1+3(2)^2-1=11\), \(P_3=11+3(3)^2-1=37\), and \(P_4=37+3(4)^2-1=84\). Hence, the correct answer is \(84\). The option \(81\) may result from incorrectly using \(3n^2-2\) instead of \(3n^2-1\). Exam tip: evaluate \(3n^2-1\) for each value of \(n\) before adding it to the preceding term.