Recursive rule

The governing concept is a recursive rule with an index-dependent decrement. To move from aₙ to aₙ₊₁, calculate the complete quantity n(n + 4) + 2 and subtract it from the current term. For n = 1, the decrement is 1(1 + 4) + 2 = 7, so a₂ = 200 − 7 = 193. For n = 2, the decrement is 2(2 + 4) + 2 = 14, giving a₃ = 193 − 14 = 179. For n = 3, it is 3(3 + 4) + 2 = 23, so a₄ = 179 − 23 = 156. Hence option A is correct. Omitting the final 2, using the wrong value of n, or subtracting only the product would produce different distractor values.

Apply the recursive rule successively with the appropriate value of n. We get a_2=2(4)+4(1)-1=11, a_3=2(11)+4(2)-1=29, and a_4=2(29)+4(3)-1=69. Therefore, the correct answer is 69. A value such as 71 can result from using an incorrect value of n in the term 4n-1. Exam tip: to find a_4, apply the rule in order for n=1, 2, and 3.

Option A gives both an initial term, \(a_1=2\), and a rule for obtaining each next term, so it determines one unique sequence. For example, \(a_2=2+3=5\). Exam tip: a recursive definition needs a starting value.

Apply the recursive rule first with \(n=1\): \(a_2=\frac{96}{2}+1^2+1=48+2=50\). Next, use \(n=2\): \(a_3=\frac{50}{2}+2^2+2=25+6=31\). Hence, 31 is correct. Option 30 may result from adding \(2^2+2\) incorrectly in the second step. Exam tip: To find \(a_3\), apply the rule successively for \(n=1\) and \(n=2\).

Use the recursive rule step by step. For \(n=1\), \(a_2=\frac{24}{3}+1^2=8+1=9\). Then, for \(n=2\), \(a_3=\frac{9}{3}+2^2=3+4=7\). Therefore, the correct answer is 7. Option 9 is the value of \(a_2\), not \(a_3\). Exam tip: substitute successive values of \(n\) in order when applying a recursive rule.

Use the recursive rule step by step. First, a_2=10+2(10)+1^2=31. Then, with n=2, a_3=31+2(10)+2^2=31+20+4=55. Hence, the correct answer is 55. The value 56 may result from incorrectly evaluating 2^2 as 5. Exam tip: substitute the correct value of n at each step, while a_1=10 remains fixed.

The recursive rule requires adding the previous two terms and 3n for the current index. Thus, a_3=6+2+3(3)=17, a_4=17+6+3(4)=35, and a_5=35+17+3(5)=67. Therefore, 67 is correct. An answer such as 62 can result from an error while adding 3(5)=15. Exam tip: find a_3 and a_4 in order before calculating a_5.

Using the recursive rule with n=3, \(a_3=a_2+a_1+3(3)=6+2+9=17\). Then, for n=4, \(a_4=a_3+a_2+3(4)=17+6+12=35\). Therefore, the correct answer is 35. A value such as 33 can result from calculating the \(3n\) term incorrectly. Exam tip: always substitute the current value of n when finding each new term.

Apply the recursive rule one step at a time. First, \(a_2=3^2-2(3)+3=9-6+3=6\). Then \(a_3=6^2-2(6)+3=36-12+3=27\). Therefore, the correct answer is 27. The value 36 is only \(6^2\); the remaining terms \(-2a_n+3\) in the rule must also be included. Exam tip: substitute the complete previous term into the rule before finding the next term.

To find the next term, put n=1 in the recursive rule: a_2=a_1^2-a_1+1. Therefore, a_2=4^2-4+1=16-4+1=13. Option 15 results from an incorrect calculation rather than from the given rule. Exam tip: for a_2, substitute a_1 directly for a_n.

(74-11=63), and (63\div7=9) steps, so the term number is (10). Exam tip: add (1) to the number of steps.

For (a_{12}), (8) is added (11) times, so (9+88=97). Exam tip: the rule is applied one less time than the term number.

Given \(a_1=4\), we get \(a_2=2(4)+7=15\), \(a_3=2(15)+7=37\), and \(a_4=2(37)+7=81\). Therefore, \(a_4-a_2=81-15=66\). Hence, 66 is the correct answer. A value such as 62 results from an error while generating the terms or subtracting them. Exam tip: In a recursive sequence, calculate the required terms in order before finding their difference.

Given a_1=4, the recursive rule gives a_2=2(4)+7=15 and a_3=2(15)+7=37. Therefore, a_3-a_2=37-15=22. The value 20 would result from an error while calculating the terms or subtracting them. Exam tip: in recursive-sequence questions, calculate the required terms in order before finding their difference.

The recursive rule says that each new term is obtained by adding the previous two terms and 6. Thus, a_3=4+8+6=18, a_4=8+18+6=32, and a_5=18+32+6=56. Therefore, 56 is correct. A value such as 54 results from an error in using the previous terms or adding 6. Exam tip: write each intermediate term in order when solving a recursive sequence.

The recursive rule forms each new term by adding the previous two terms and 6. Thus, \(a_3=8+4+6=18\), and \(a_4=18+8+6=32\). Therefore, the correct answer is 32. The value 26 is only \(a_2+a_3\); it misses the required addition of 6. Exam tip: find \(a_3\) before calculating \(a_4\), and include the constant at every step.

In an arithmetic progression, the difference between consecutive terms is constant. Here, \(a_{n+1}-a_n=4\), so every term increases by 4. In option C, the difference is \(4n\), which changes. Exam tip: test an AP rule using \(a_{n+1}-a_n\).

Apply the recursive rule successively for n=1, 2, and 3: a_2=5+(8×1-3)=10, a_3=10+(8×2-3)=23, and a_4=23+(8×3-3)=44. Therefore, the correct answer is 44. A value such as 41 results from using an incorrect increment at one of the steps. Exam tip: to find a_4, apply the rule step by step from n=1 to n=3.

Putting n=1 gives a_2=a_1+8(1)-3=5+5=10. Then, putting n=2 gives a_3=a_2+8(2)-3=10+13=23. Therefore, the correct answer is 23. A value such as 22 may result from incorrectly treating the increment as constant, whereas 8n-3 changes with n. Exam tip: in a recursive relation, substitute the correct value of n at every step.

The original options did not contain the mathematically correct value, so the options have been corrected. Apply the recursive rule for n = 1, 2, and 3 because a₄ requires three transitions. First, a₂ = 180 − (6×1 + 5) = 180 − 11 = 169. Next, a₃ = 169 − (6×2 + 5) = 169 − 17 = 152. Finally, a₄ = 152 − (6×3 + 5) = 152 − 23 = 129. Therefore option A, 129, is correct. The subtractions are 11, 17, and 23, increasing by 6. The previously supplied value 136 would result from an arithmetic or indexing mistake and is not compatible with the stated recurrence.