Recursive rule

Correct answer: A. 55

Explanation: Each new term is obtained from the previous two terms using the recursive rule. Q_3=5+2(2)+3=12, then Q_4=12+2(5)+4=26, and Q_5=26+2(12)+5=55. Hence, the correct answer is 55. A value such as 57 can result from incorrectly handling 2Q_{n-2} or the added n. Exam tip: substitute n=3, 4, and 5 step by step.

Correct answer: C. 12

Explanation: In a recursive rule, first find each next term from the preceding term. R_2=R_1^2+2=1^2+2=3. Then R_3=R_2^2+3=3^2+3=12. Therefore, 12 is correct. A value such as 10 can result from not adding the correct term number after squaring R_2. Exam tip: write the new value of n at every step.

Correct answer: B. 71

Explanation: The recursive rule gives each term from the preceding term. Thus, \(S_2=100-2^2=96\), \(S_3=96-3^2=87\), and \(S_4=87-4^2=87-16=71\). Therefore, 71 is correct. A value such as 69 can result from using incorrect squares in the steps. Exam tip: since the sequence starts at \(S_1\), subtract \(2^2,3^2\), and \(4^2\) to find \(S_4\), not \(1^2\).

Correct answer: D. 61

Explanation: In the recursive rule, substitute the index n for each term being calculated. Thus, \(T_2=3\times2-2=4\), \(T_3=4\times4-3=13\), and \(T_4=5\times13-4=61\). Therefore, the correct answer is \(61\). A close option such as \(59\) can result from an error in multiplying by 5 or subtracting 4 in the final step. Exam tip: To find \(T_4\), calculate \(T_2\) and \(T_3\) in order first.

Correct answer: A. \(a_1=3,\ a_n=a_{n-1}+4\ (n\ge2)\)

Explanation: A recursive definition needs an initial term and a rule to generate each next term. In A, \(a_1\) is given, so \(a_2=3+4=7\). B lacks an initial term, while D also needs \(a_2\). Exam tip: always check that all required starting terms are provided.

Correct answer: A. 10

Explanation: Using the recursive rule, \(a_3=a_2+3(3)=a_2+9\) and \(a_4=a_3+3(4)=a_3+12\). Hence, \(31=a_2+9+12=a_2+21\), so \(a_2=10\). If \(a_2=12\), then \(a_4\) would be 33, not 31. Exam tip: When a later term is given, expand the recurrence backwards to reach the required term.

Correct answer: B. 3

Explanation: Using the recursive rule, \(b_2=2b_1+2\). Hence, \(b_3=2b_2+3=2(2b_1+2)+3=4b_1+7\). Since \(b_3=19\), we get \(4b_1+7=19\), so \(b_1=3\). If 2 is used instead, it gives \(b_3=15\), so it is not correct. Exam tip: while expanding a recurrence, do not forget to add the value of the index at each step.

Correct answer: D. \(c_2=6\) and \(c_3=15\)

Explanation: Using the recursive rule with \(n=2\), \(c_2=c_1+2^2=2+4=6\). Then, for \(n=3\), \(c_3=c_2+3^2=6+9=15\). Therefore, option D is correct. In option C, \(c_2\) is correct, but \(3^2=9\) has not been added correctly to find \(c_3\). Exam tip: to obtain each new term, add the square of the current index to the previous term.

Correct answer: C. (5)

Explanation: (d_4=28) and (d_5=43), so the condition is first satisfied at (n=5). In exams, choose the first term that crosses the limit.

Correct answer: A. 47

Explanation: Using the recursive rule, \(e_2=2(3)+2=8\), \(e_3=2(8)+3=19\), \(e_4=2(19)+4=42\), and \(e_5=2(42)+5=89\). Therefore, \(e_5-e_4=89-42=47\). The value 49 does not result because finding \(e_5\) requires using \(2e_4\). Exam tip: In a recursive sequence, calculate the required terms one by one in order.

Correct answer: D. (40)

Explanation: The direct answer is Option D: 40. Use the recurrence carefully, beginning with f₁=2 and applying fₙ=fₙ₋₁+2n. For n=2, f₂=f₁+2(2)=2+4=6. For n=3, f₃=f₂+2(3)=6+6=12. For n=4, f₄=f₃+2(4)=12+8=20. The first four terms are therefore 2, 6, 12 and 20. Their sum is 2+6+12+20=40. Option A, 28, is wrong because it does not equal the required total. Option B, 32, is also too small and results from an arithmetic or term-copying mistake. Option C, 36, is close but still omits or miscalculates part of the total. Option D is correct because it adds all four correctly generated terms. The important idea is that a recursive formula does not give every term directly; each new term depends on the previous one. Also, the question asks for a sum, so finding only f₄ is not enough. Exam cue: write every term in a column, check the multiplication 2n, and add all requested terms only after generating them.

Correct answer: B. \(14:5\)

Explanation: Using the recursive rule, \(g_2=3(1)-1=2\), \(g_3=3(2)-1=5\), and \(g_4=3(5)-1=14\). Hence, \(g_4:g_3=14:5\), so option B is correct. A ratio such as \(12:5\) results from using an incorrect term value. Exam tip: calculate the terms in order up to the required term and keep the ratio in the stated order \(g_4:g_3\).

Correct answer: C. 40

Explanation: Apply the recursive rule successively: \(h_3=2(7)-2+3=15\), \(h_4=2(15)-7+3=26\), and \(h_5=2(26)-15+3=40\). Therefore, the correct answer is 40. A value such as 38 can result from mishandling the signs of the previous terms or the \(+3\) term. Exam tip: before substituting, write down \(h_{n-1}\) and \(h_{n-2}\) separately for each new term.

Correct answer: A. 41

Explanation: In the recursive rule, add n^2-n to the preceding term for each value of n. Thus, p_2=1+(2^2-2)=3, p_3=3+(3^2-3)=9, p_4=9+(4^2-4)=21, and p_5=21+(5^2-5)=41. Therefore, the correct answer is 41. A value such as 43 can result from incorrectly handling the first increment, 2. Exam tip: To find p_5, list the successive values from n=2 to n=5 systematically.

Correct answer: C. (135)

Explanation: The direct answer is C, 135. A recursive rule finds each term from earlier terms. Start with a_1=4 and use a_n=2a_{n-1}+n+2, where n is the index of the term being found. For n=2: a_2=2(4)+2+2=12. For n=3: a_3=2(12)+3+2=29. For n=4: a_4=2(29)+4+2=64. For n=5: a_5=2(64)+5+2=135. Option A, 129, results from an arithmetic mistake. Option B, 132, does not satisfy the final calculation. Option C is correct because it uses the previous term 64 and the current index 5. Option D, 138, also comes from adding incorrectly. Exam cue: in a recurrence, use the current n in the extra term.

Correct answer: B. 140

Explanation: The governing concept is a recursive rule: each term is calculated from the preceding term, and the value of n must be substituted separately at every step. Start with b₁ = 7. For n = 2, b₂ = 3(7) − 2(2) + 1 = 21 − 4 + 1 = 18. For n = 3, b₃ = 3(18) − 6 + 1 = 49. For n = 4, b₄ = 3(49) − 8 + 1 = 147 − 7 = 140. Thus option B is correct. Option A can result from an arithmetic slip in the final subtraction, while 144 and 148 do not follow the stated recurrence. It is important not to treat the expression as a constant-difference sequence.

Correct answer: D. 55

Explanation: Use the recursive rule step by step: \(c_3=5+2(2)+3=12\), \(c_4=12+2(5)+4=26\), and \(c_5=26+2(12)+5=55\). Therefore, the correct answer is \(55\). A close option such as \(53\) may result from not including the term \(2c_{n-2}\) correctly. Exam tip: write the previous two terms and the current value of \(n\) separately before substituting.

Correct answer: A. 13

Explanation: The governing concept is a recursive sequence evaluated backward. The recurrence adds 5n − 1 when moving from dₙ₋₁ to dₙ. Thus the increments are 9 for n = 2, 14 for n = 3, 19 for n = 4 and 24 for n = 5. Starting from d₁, d₅ = d₁ + 9 + 14 + 19 + 24 = d₁ + 66. Since d₅ = 79, d₁ = 79 − 66 = 13. Therefore option A is correct. A useful check is to rebuild the sequence forward: 13 + 9 = 22, then +14 = 36, then +19 = 55, then +24 = 79. The other choices arise if one increment is omitted, calculated with the wrong n, or subtracted incorrectly.

Correct answer: D. Rule (u_n=2u_{n-1}+2)

Explanation: Each new term is obtained by doubling the previous term and adding (2). In exams, test the rule at two consecutive places.

Correct answer: C. (6)

Explanation: (f_5=109) and (f_6=221), so the condition is first satisfied at (n=6). In exams, choose the first term that crosses the limit.

Correct answer: C. −4

Explanation: The governing concept is a recursive rule with an alternating sign. At each step, calculate (−1)ⁿ first: it is positive for even n and negative for odd n. Starting with g₁ = 10, for n = 2 the added amount is (+1)(2²) = 4, so g₂ = 14. For n = 3 the amount is (−1)(3²) = −9, so g₃ = 5. For n = 4, add 16 to obtain g₄ = 21. For n = 5, subtract 25, giving g₅ = 21 − 25 = −4. Hence option C is correct. The distractors reflect sign errors or failure to square n; the alternating factor must be applied before adding each term.

Correct answer: A. 42

Explanation: In a recursive rule, each term is calculated from the preceding term. First, \(h_2=(2-1)\times2+3=5\). Then \(h_3=(3-1)\times5+3=13\), and \(h_4=(4-1)\times13+3=42\). Therefore, the correct answer is 42. A value such as 45 may result from incorrectly using \(n\) instead of \(n-1\), or from using the wrong multiplier. Exam tip: substitute the value of \(n\) first and then use the previous term carefully.

Correct answer: A. One initial term and the recursive rule

Explanation: In a first-order rule, each new term depends only on the preceding term. Thus, one initial term plus the rule fixes every later term. A rule alone can produce many sequences. Exam tip: the number of initial values matches the order of recurrence.

Correct answer: D. (13)

Explanation: The direct answer is Option D: 13. The recurrence is kₙ=kₙ₋₁/2+n, so first halve the previous term and then add the current index n. Begin with k₁=80. For n=2: k₂=80/2+2=40+2=42. For n=3: k₃=42/2+3=21+3=24. For n=4: k₄=24/2+4=12+4=16. For n=5: k₅=16/2+5=8+5=13. Therefore Option D is correct. Option A, 10, is wrong because it omits or mishandles the final addition. Option B, 11, is wrong for the same calculation reason. Option C, 12, is the halved value before adding n at the last step, so it shows stopping too early. Option D correctly performs both operations in the stated order. Parentheses matter: the formula means half of the previous term, then add n, not half of the whole expression or add a different index. Exam cue: write the index beside every line and use “divide first, add n second.”

Correct answer: A. \(59\)

Explanation: Apply the recursive rule step by step: \(m_2=3(2)+2-2=6\), \(m_3=3(6)+3-2=19\), and \(m_4=3(19)+4-2=59\). Hence, the correct value is \(59\). A close option such as \(61\) may result from adding the \(n-2\) term incorrectly. Exam tip: write the value of \(n\) separately at every step before substituting.