Recursive rule

Correct answer: D. 14

Explanation: In a recursive rule, each new term is calculated from the preceding term. \(k_2=\frac{64}{2}+2=34\), \(k_3=\frac{34}{2}+3=20\), and \(k_4=\frac{20}{2}+4=14\). Hence, the correct answer is \(14\). A choice such as \(12\) can result from missing the division by 2 before adding 4. Exam tip: at every step, first divide the previous term by 2 and then add the current index \(n\).

Correct answer: A. 29

Explanation: In the recursive rule, substitute the current index each time. Thus, \(m_2=2+3(2)=8\), \(m_3=8+3(3)=17\), and \(m_4=17+3(4)=29\). Therefore, 29 is correct. A value such as 32 would result from using an incorrect increment in the final step instead of \(3n\). Exam tip: when \(m_1\) is given, begin finding the next term with \(n=2\).

Correct answer: C. (-10)

Explanation: The recurrence is \(n_n=2n_{n-1}-n^2\), with \(n_1=5\). To find the second term, substitute \(n=2\): \(n_2=2n_1-2^2=2(5)-4=6\). For the third term, substitute \(n=3\): \(n_3=2n_2-3^2=2(6)-9=3\). For the fourth term, substitute \(n=4\): \(n_4=2n_3-4^2=2(3)-16=6-16=-10\). Thus option C is correct.

The symbol on the left identifies the term being calculated, while the index in \(n^2\) changes at every step. One must not use the same square throughout the recurrence. The sequence begins \(5,6,3,-10\), so a negative fourth term is expected and is not a reason to stop. Option C, negative ten, matches the exact calculation; the other choices result from an error in the subtraction or in selecting the index.

Correct answer: B. 26

Explanation: By the recursive rule, each new term equals the previous term plus 3 times the term two places before it. Thus, \(r_3=2+3(1)=5\), \(r_4=5+3(2)=11\), and \(r_5=11+3(5)=26\). Therefore, the correct answer is 26. A value such as 23 can result from not multiplying the second previous term by 3 correctly. Exam tip: list the terms in order and identify \(r_{n-1}\) and \(r_{n-2}\) separately at each step.

Correct answer: A. 10

Explanation: Using the recursive rule, \(t_2=x+2\), \(t_3=x+2+3\), \(t_4=x+2+3+4\), and \(t_5=x+2+3+4+5=x+14\). Since \(t_5=24\), we get \(x+14=24\), so \(x=10\). If \(x=11\), then \(t_5=25\), so it is not correct. Exam tip: In a recursive sequence, list every added term up to the required term before solving for the first term.

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

Explanation: The correct rule is \(a_1=2\) and \(a_n=3a_{n-1}\), because every term is three times the preceding term: \(2\times3=6\), \(6\times3=18\), and \(18\times3=54\). Option B gives the second term as 6, but its next term would be \(2\times6+2=14\), not 18. Exam tip: when ratios of consecutive terms are equal, check for a multiplicative recursive rule.

Correct answer: D. 49

Explanation: In the recursive rule, multiply the preceding term by 2 and then subtract 1. Thus, \(v_2=2(4)-1=7\), \(v_3=2(7)-1=13\), \(v_4=2(13)-1=25\), and \(v_5=2(25)-1=49\). Therefore, the correct answer is 49. Although 47 is a close distractor, it is not obtained by applying the given rule successively. Exam tip: at each step, double the previous term before subtracting 1.

Correct answer: B. 30

Explanation: Apply the recursive rule step by step: \(w_2=0+2^1=2\), \(w_3=2+2^2=6\), \(w_4=6+2^3=14\), and \(w_5=14+2^4=30\). Therefore, the correct answer is 30. A value such as 28 can result from using the final increment incorrectly. Exam tip: from \(w_1\) to \(w_5\), apply the rule four times.

Correct answer: C. 21

Explanation: In a recursive rule, first find the preceding term. \(x_2=2x_1+2^2=2(1)+4=6\). Then \(x_3=2x_2+3^2=2(6)+9=21\). Therefore, the correct answer is 21. A value such as 18 can result from incorrectly using \(x_1\) in place of \(x_2\). Exam tip: substitute the immediately preceding term at each step.

Correct answer: A. 37

Explanation: Apply the recursive rule step by step: (y_2=7+1^2=8), (y_3=8+2^2=12), (y_4=12+3^2=21), and (y_5=21+4^2=37). Therefore, the correct answer is 37. A value such as 39 can result from adding the squares or the initial term incorrectly. Exam tip: to find (y_5), include exactly the increments (1^2,2^2,3^2,4^2).

Correct answer: D. 155

Explanation: By the recursive rule, each new term is obtained by multiplying the previous two terms and adding 1. Thus, z_3=3×2+1=7, z_4=7×3+1=22, and z_5=22×7+1=155. Therefore, the correct answer is 155. A value such as 153 can result from forgetting the final “+1” in the rule. Exam tip: write each term in sequence and use both preceding terms at every step.

Correct answer: B. 41

Explanation: A recursive rule uses the preceding term to find each new term. \(A_2=2\times1+1=3\), \(A_3=3\times3+1=10\), and \(A_4=4\times10+1=41\). Hence, the correct answer is 41. An option such as 37 results from not applying the multiplication by 4 correctly while finding \(A_4\). Exam tip: write the value of \(n\) first, then multiply the previous term by it and add 1.

Correct answer: C. 37

Explanation: In the recursive rule, each term is obtained by adding the relevant power to the preceding term. Thus, \(B_2=9+2^2=13\), \(B_3=13+2^3=21\), and \(B_4=21+2^4=37\). Therefore, 37 is correct. Getting 39 would result from using an incorrect exponent or starting the additions incorrectly. Exam tip: when \(B_1\) is given, use \(n=2\) in the rule to find the next term.

Correct answer: A. 25

Explanation: Apply the recursive rule step by step: \(C_2=5+2(2-1)=7\), \(C_3=7+3(3-1)=13\), and \(C_4=13+4(4-1)=25\). Hence, the correct answer is \(25\). A close distractor such as \(27\) can result from adding \(4(4-1)=12\) incorrectly. Exam tip: calculate \(n(n-1)\) first for each term, then add it to the preceding term.

Correct answer: D. 27

Explanation: Each term is found from the previous term using the recursive rule. Thus, \(D_2=2\times2+1=5\), \(D_3=2\times5+2=12\), and \(D_4=2\times12+3=27\). Therefore, the correct answer is 27. An answer such as 25 can result from adding the \(n-1\) term incorrectly. Exam tip: To find \(D_4\), calculate \(D_2\) and \(D_3\) in order first.

Correct answer: C. 22

Explanation: In the recursive rule, each new term is obtained by adding 2 to the sum of the two immediately preceding terms. Thus, E_3=1+4+2=7, E_4=4+7+2=13, and E_5=7+13+2=22. Therefore, the correct answer is 22. The value 20 can result from not using the correct two previous terms in the final step. Exam tip: Write the terms one by one, starting from E_3.

Correct answer: A. 42

Explanation: In a recursive rule, each new term is found from the preceding term. Thus, \(G_2=2+2(3)=8\), \(G_3=3+2(8)=19\), and \(G_4=4+2(19)=42\). Therefore, the correct answer is 42. A value such as 44 can result from using an incorrect previous term. Exam tip: to find \(G_4\), calculate \(G_2\) and \(G_3\) in order first.

Correct answer: D. 40

Explanation: In a recursive rule, each term is found from the previous term. \(H_2=2+2(3)=8\), \(H_3=8+3(4)=20\), and \(H_4=20+4(5)=40\). Hence, the correct answer is \(40\). A value such as \(38\) can result from using an incorrect increase instead of \(4(5)\) in the final step. Exam tip: Start with \(H_1\) and calculate the terms successively for \(n=2,3,4\).

Correct answer: C. 20

Explanation: Apply the recursive rule using the current value of n: J_2=50-3(1)=47, J_3=47-3(2)=41, J_4=41-3(3)=32, and J_5=32-3(4)=20. Therefore, the correct answer is 20. The value 23 may result from subtracting only 3 in the last step instead of 3(n-1), which does not follow the rule. Exam tip: To move from J_1 to J_5, write the four steps for n=2,3,4,5.

Correct answer: B. (17)

Explanation: The direct answer is B: 17. The recurrence is \\(K_n=2K_{n-1}+K_{n-2}\\), so each new term is twice the immediately previous term plus the term before it. Start with \\(K_1=1\\) and \\(K_2=1\\). For n=3, \\(K_3=2K_2+K_1=2(1)+1=3\\). For n=4, \\(K_4=2K_3+K_2=2(3)+1=7\\). For n=5, \\(K_5=2K_4+K_3=2(7)+3=17\\). Therefore option B is correct. Option A, 15, does not result from the rule and may come from an addition or doubling error. Option C, 19, is not obtained when the correct earlier terms are used. Option D, 21, is also inconsistent with the recurrence. The common mistake is to double both earlier terms or to use only the previous term; the rule specifically says double the previous term and then add the term before it. Memory cue: “double the neighbour, add the older one.”

Correct answer: A. 38

Explanation: Each new term is obtained by adding \(2n+1\) to the preceding term. Thus, \(L_2=6+5=11\), \(L_3=11+7=18\), \(L_4=18+9=27\), and \(L_5=27+11=38\). Therefore, 38 is correct. A value such as 40 results from using an incorrect increment at one of the steps. Exam tip: in a recursive rule, substitute consecutive values of \(n\), starting with the second term.

Correct answer: D. 18

Explanation: For a recursive rule, first calculate the preceding term. \(M_2=3M_1+2=3\times1+2=5\). Then \(M_3=3M_2+3=3\times5+3=18\). Therefore, 18 is correct. An answer such as 16 can result from using an incorrect value of \(M_2\) or of \(n\). Exam tip: remember to add the current index \(n\) at every step.

Correct answer: C. 88

Explanation: A recursive rule uses the preceding term to obtain the next term. \(R_2=2\times2+2=6\), \(R_3=3\times6+3=21\), and \(R_4=4\times21+4=88\). Hence, 88 is correct. A value such as 82 can result from an error while adding \(4\) in the final step. Exam tip: substitute the current value of \(n\) separately at every step.

Correct answer: B. 41

Explanation: Under the recursive rule, each new term is obtained by adding \(2^n+n\) to the previous term. \(O_2=4+2^2+2=10\), \(O_3=10+2^3+3=21\), and \(O_4=21+2^4+4=41\). Therefore, the correct answer is 41. A value such as 45 may result from incorrectly adding an extra 4 while finding the fourth term. Exam tip: write the value of \(n\) separately at every step before evaluating the power.

Correct answer: D. 47

Explanation: Substituting n=2,3,4,5 in the recursive rule gives increments of 5, 9, 13, and 17. Thus, P_2=8, P_3=17, P_4=30, and P_5=47. Therefore, 47 is correct. A value such as 45 may result from an incorrect final increment. Exam tip: in a recursive sequence, calculate each term successively from the preceding term.