Recursive rule

Correct answer: C. 70

Explanation: Direct answer: C, 70. The rule is recursive because each new term is calculated from the two preceding terms. Start with a₁=2 and a₂=5. For n=3, a₃=2a₂+a₁=2(5)+2=12. For n=4, use the two previous terms again: a₄=2a₃+a₂=2(12)+5=29. For n=5, a₅=2a₄+a₃=2(29)+12=70. Therefore C is correct. A, 62, is not produced by the stated recurrence and may result from an arithmetic error. B, 66, also does not follow from 2(29)+12. C, 70, uses both required preceding terms and the coefficient 2 correctly. D, 74, is another result of incorrect addition or indexing. The complete beginning of the sequence is 2, 5, 12, 29, 70. Common warning: aₙ₋₂ means the second previous term; omitting it changes the rule completely.

Correct answer: C. 63

Explanation: This is a second-order recursive rule, so each new term uses the two preceding terms. Calculate a₃ = 3(7) − 2(3) = 21 − 6 = 15. Then a₄ = 3(15) − 2(7) = 45 − 14 = 31. Finally, a₅ = 3(31) − 2(15) = 93 − 30 = 63. Thus option C is correct; using only one previous term would not follow the stated rule.

Correct answer: C. 32

Explanation: Apply the recursive rule successively for each value of n. \(a_2=4+2^1+1^2=7\), \(a_3=7+2^2+2^2=15\), and \(a_4=15+2^3+3^2=15+8+9=32\). Therefore, the correct answer is 32. A value such as 29 can result from adding \(3^2\) incorrectly. Exam tip: to find \(a_4\), substitute \(n=1,2,3\) in order.

Correct answer: C. 80

Explanation: Using n=1,2,3,4 in the recursive rule, the values subtracted are 3, 6, 11, and 20. Hence, a_5=120-(3+6+11+20)=120-40=80. The value 82 is obtained after only the first three subtractions, so it is a_4, not a_5. Exam tip: to find a_5 from a_1, apply the rule four times.

Correct answer: B. 8

Explanation: Direct answer: B, 8. Apply the recurrence four times because a₁ must be advanced to a₅. The factor (−1)ⁿ⁺¹ alternates: it is positive for even powers and negative for odd powers. For n=1, a₂=10+(−1)²(3)=13. For n=2, a₃=13+(−1)³(4)=9. For n=3, a₄=9+(−1)⁴(5)=14. For n=4, a₅=14+(−1)⁵(6)=8. Thus B is correct. A, 6, would require an incorrect extra subtraction or index. C, 10, ignores the accumulated changes and simply returns the starting value. D, 12, results from mishandling one alternating sign or one increment. The sequence is 10, 13, 9, 14, 8. A reliable memory cue is: (−1) to an even power equals +1, while (−1) to an odd power equals −1; always evaluate the exponent before applying the sign.

Correct answer: C. (34)

Explanation: This rule builds the sequence by adding a quantity that depends on n. Since the required term is a₄ and the sequence starts at a₁, the rule must be used three times, with n equal to 1, 2, and 3. The added amount is always n(n + 2) + 1, so the multiplication and the final addition must both be performed at each step.

For n = 1, the added amount is 1(3) + 1 = 4, giving a₂ = 5 + 4 = 9. For n = 2, it is 2(4) + 1 = 9, giving a₃ = 18. For n = 3, it is 3(5) + 1 = 16, giving a₄ = 34. Thus the successive additions are 4, 9, and 16, whose sum is 29; 5 + 29 also equals 34. Therefore option C is correct. Keeping n matched to the step prevents using the wrong expression.

Correct answer: C. 121

Explanation: Use n = 1, 2, and 3 successively because these produce a₂, a₃, and a₄. The amounts subtracted are 1(3)+1 = 4, 2(4)+1 = 9, and 3(5)+1 = 16. Therefore a₂ = 150−4 = 146, a₃ = 146−9 = 137, and a₄ = 137−16 = 121. Option C is correct; subtracting only n(n+2) would omit the final −1.

Correct answer: C. 42

Explanation: Use the current value of n at each step of the recursive rule. a_2=2(2)+3(1)-1=6, a_3=2(6)+3(2)-1=17, and a_4=2(17)+3(3)-1=42. Therefore, 42 is correct. A value such as 39 can result from an error while calculating 2a_3 in the final step. Exam tip: explicitly write n=1, 2, and 3 while finding a_2, a_3, and a_4 respectively.

Correct answer: B. \(\frac{41}{2}\)

Explanation: The recurrence is \(a_{n+1}=\frac{a_n}{2}+n^2\), with initial value \(a_1=64\). To find \(a_2\), use \(n=1\): \(a_2=\frac{64}{2}+1^2=32+1=33\). To find \(a_3\), use \(n=2\): \(a_3=\frac{33}{2}+2^2=\frac{33}{2}+4=\frac{33}{2}+\frac{8}{2}=\frac{41}{2}\). Hence option B is correct.

The index must change from 1 to 2 because the first application produces the second term and the second application produces the third term. Keeping the fraction exact avoids rounding and gives the precise result. The decimal value would be 20.5, but \(\frac{41}{2}\) is the exact form requested by the options. The other fractional choices differ by 1 from this result and arise from an arithmetic error in the final addition.

Correct answer: C. 6

Explanation: Direct answer: C, 6. Because the rule is recursive, calculate the terms in order. Begin with a₁=12. Put n=1 into aₙ₊₁=aₙ/3+2n: a₂=a₁/3+2(1)=12/3+2=4+2=6. Now put n=2 and use the value just found: a₃=a₂/3+2(2)=6/3+4=2+4=6. Therefore C is correct. A, 4, is only the divided part of the calculation and omits the added 2n term. B, 5, comes from an arithmetic or indexing error. C, 6, follows from both steps and both parts of the recurrence. D, 7, likewise does not result from substituting n=2 correctly. The important idea is that a₃ depends on a₂, so it cannot be found directly from a₁ without first calculating a₂. Keep the index attached to the correct transition: n=1 creates a₂, while n=2 creates a₃.

Correct answer: C. 29

Explanation: The governing concept is an index-dependent second-order recurrence. Each term is obtained by adding the two preceding terms and then adding the current index n. Beginning with a₁ = 1 and a₂ = 4, use n = 3 to get a₃ = a₂ + a₁ + 3 = 4 + 1 + 3 = 8. Next, a₄ = a₃ + a₂ + 4 = 8 + 4 + 4 = 16. Finally, for n = 5, a₅ = a₄ + a₃ + 5 = 16 + 8 + 5 = 29. Therefore option C is correct. The added index cannot be omitted: without it the values would be smaller. Options A, B, and D can result from using the wrong preceding terms, forgetting one index contribution, or making an addition error.

Correct answer: C. 57

Explanation: In a recursive rule, each new term is calculated from the preceding term. First, \(a_2=2^2+2+1=7\). Then \(a_3=7^2+7+1=49+7+1=57\). Therefore, the correct answer is 57. A value such as 59 usually results from an error while squaring or adding. Exam tip: write the previous term separately before calculating the next term.

Correct answer: C. 96

Explanation: In a recursive rule, each new term is calculated from the preceding term. First, \(a_2=6^2-4(6)=36-24=12\). Then \(a_3=12^2-4(12)=144-48=96\). Therefore, the correct answer is 96. A value such as 84 can result from an error in the second calculation. Exam tip: always find \(a_2\) before calculating \(a_3\).

Correct answer: C. (12)th

Explanation: (53-9=44), and (44\div4=11) steps, so the term number is (12). Exam tip: add (1) to the number of steps.

Correct answer: C. (61)

Explanation: For (a_{10}), (6) is added (9) times, so (7+54=61). Exam tip: the rule is applied one less time than the term number.

Correct answer: C. 48

Explanation: Using the recursive rule, \(a_2=2(3)+5=11\), \(a_3=2(11)+5=27\), and \(a_4=2(27)+5=59\). Therefore, \(a_4-a_2=59-11=48\), so option C is correct. A value such as \(45\) can result from an error in finding an intermediate term or subtracting. Exam tip: write the required terms in order before taking their difference.

Correct answer: C. 42

Explanation: The recursive rule says that each new term is the sum of the previous two terms plus 5. Thus, \(a_3=6+2+5=13\), \(a_4=13+6+5=24\), and \(a_5=24+13+5=42\). Therefore, 42 is correct. A value such as 46 results from using the preceding terms or the added constant incorrectly. Exam tip: at every step, include exactly the previous two terms and the +5.

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

Explanation: In \(a_{n+1}=-2a_n\), the factor 2 doubles the magnitude and the minus sign reverses the sign. \(2a_n\) does not reverse it. Exam tip: check a multiplier’s sign and magnitude separately.

Correct answer: A. (a_1=2, a_{n+1}=2a_n+n^2)

Explanation: A recursive rule must reproduce the sequence from its first term. For option A, the initial value is \(a_1=2\), and the next-term rule is \(a_{n+1}=2a_n+n^2\). To test it, use \(n=1\) to produce the second term and \(n=2\) to produce the third term. Matching successive terms is the key check.

Using option A, \(a_2=2(2)+1^2=5\), and \(a_3=2(5)+2^2=14\). Continuing, \(a_4=2(14)+3^2=37\), exactly as required. Option B would give 11 for the third term, option C starts with the wrong first term, and option D gives 5 followed by 14? Actually it gives \(a_3=14\) but \(a_4=41\), so it fails later. Therefore A is correct.

Correct answer: B. a₁ = 6, aₙ₊₁ = aₙ + 2n + 1

Explanation: A recursive rule must specify the correct starting term and reproduce every successive term from the previous one. The sequence begins with 6, and its successive differences are 9−6 = 3, 14−9 = 5, and 21−14 = 7. In the rule aₙ₊₁ = aₙ + 2n + 1, the added amounts are 3 for n = 1, 5 for n = 2, and 7 for n = 3. Starting from a₁ = 6, it therefore produces 9, 14, and 21 exactly. Thus option B is correct. Option A adds 2, 4, and 6, so it produces a different sequence. Option C has the wrong first term, while option D produces 6, 9, 15, 27. Checking both the initial value and the transition confirms the answer.

Correct answer: C. 48

Explanation: The recursive rule uses the current index n at each step. Thus, a_2=1×3+2=5, then a_3=2×5+4=14, and a_4=3×14+6=48. Therefore, the correct answer is 48. A value such as 45 may result from using an incorrect index in the final step. Exam tip: to find a_4, apply the rule successively for n=1, 2, and 3.

Correct answer: C. 31

Explanation: In the recursive rule, the value of n changes at every step. For n=1, a_2=2a_1-1=2×2-1=3. Then, for n=2, a_3=3a_2-1=3×3-1=8, and for n=3, a_4=4a_3-1=4×8-1=31. Therefore, the correct answer is 31. The value 33 would result from forgetting to subtract 1 in the final step. Exam tip: write the current index n before applying a recursive rule to check the multiplier.

Correct answer: C. 172

Explanation: To obtain a₄, apply the rule for n = 1, 2, and 3. The corresponding subtractions are 2¹ + 1² = 3, 2² + 2² = 8, and 2³ + 3² = 17. Thus a₂ = 200−3 = 197, a₃ = 197−8 = 189, and a₄ = 189−17 = 172. Option C is correct. Confusing powers with products or omitting the square term leads to the distractors.

Correct answer: C. 46

Explanation: Apply the recursive rule successively for n=1, 2, and 3: \(a_2=1+3^1+1=5\), \(a_3=5+3^2+2=16\), and \(a_4=16+3^3+3=46\). Hence, 46 is correct. The value 49 would result from incorrectly adding an extra amount in the final step. Exam tip: to find \(a_4\), use the recurrence consecutively from n=1 through n=3.

Correct answer: C. (33)

Explanation: The direct answer is option C, \(a_4=33\). This is a recursive sequence, so each new term is calculated from earlier terms. We know \(a_1=2\) and \(a_2=3\), and the rule is \(a_n=2a_{n-1}+3a_{n-2}\). For \(n=3\), use \(a_2\) and \(a_1\): \(a_3=2(3)+3(2)=6+6=12\). For \(n=4\), use \(a_3\) and \(a_2\): \(a_4=2(12)+3(3)=24+9=33\). Option A, 27, does not follow the stated recurrence. Option B, 30, results from an arithmetic or coefficient error. Option C, 33, matches both the rule and the calculation. Option D, 36, is also not obtained when the correct preceding terms are used. The important point is that the coefficient 2 multiplies the immediately previous term, while coefficient 3 multiplies the term before that. Never use \(a_1\) again when calculating \(a_4\).