Recursive rule

Check the differences between consecutive terms: \(42-36=6\), \(36-30=6\), and \(30-24=6\). Thus, each new term is obtained by subtracting 6 from the preceding term: \(a_{n+1}=a_n-6\). If 5 were subtracted, the next term after 42 would be 37, which does not match the sequence. Exam tip: always compare two consecutive terms before writing a recursive rule.

Check the ratio of consecutive terms: \(20\div5=4\), \(80\div20=4\), and \(320\div80=4\). Thus, each term is obtained by multiplying the previous term by 4, so the multiplier in the recursive rule is 4. Note that 5 is the first term, not the multiplier. Exam tip: divide a term by the preceding term and verify the same ratio with the next pair.

By the recursive rule, each new term is obtained by adding 7 to the preceding term. Thus, \(a_2=21\), \(a_3=28\), and \(a_4=35\). The closest distractor, 28, is \(a_3\), not \(a_4\). Exam tip: from \(a_1\) to \(a_4\), add 7 three times.

The recursive rule adds 10 to each preceding term. Thus, \(a_2=6+10=16\) and \(a_3=16+10=26\). Therefore, the correct answer is \(26\). Note that \(16\) is \(a_2\), not \(a_3\). Exam tip: in recursive sequences, write each step up to the required term.

The recursive rule makes each new term one-fourth of the preceding term. Thus, \(a_2=\frac{96}{4}=24\) and \(a_3=\frac{24}{4}=6\). Therefore, the correct answer is 6. Option 24 is \(a_2\), not \(a_3\). Exam tip: to find \(a_3\) from \(a_1\), apply the rule twice.

The recursive rule states that each new term is the sum of the previous two terms. Thus, a_3=1+5=6, a_4=5+6=11, and a_5=6+11=17. Therefore, 17 is correct. The value 11 is the close distractor because it is a_4, not a_5. Exam tip: write the terms in order until you reach the required term.

The recursive rule adds 2 to each preceding term. Thus, a_2=11, a_3=13, and a_4=15. The value 13 is the third term, so it is a close but incorrect option. Exam tip: start from a_1 and count the term positions carefully.

The governing concept is a recursive rule: each term is obtained from the immediately preceding term. Starting with a₁ = 28, subtract 4 repeatedly: a₂ = 28 − 4 = 24, a₃ = 24 − 4 = 20, a₄ = 20 − 4 = 16, and a₅ = 16 − 4 = 12. Hence option C is correct. The same result can be obtained by recognizing an arithmetic progression with common difference d = −4: a₅ = a₁ + (5 − 1)d = 28 + 4(−4) = 12. Option A is obtained by too many subtractions, option B does not correspond to any term, and option D results from subtracting only 3 from the initial value.

By the recursive rule, each new term is twice the previous term. Thus, \(a_2=2\times6=12\), \(a_3=2\times12=24\), and \(a_4=2\times24=48\). Therefore, 48 is correct. The value 24 is \(a_3\), so it is a close but incorrect option. Exam tip: start from \(a_1\) and count the term positions carefully.

The recursive rule makes each new term half of the preceding term. Thus, \(a_2=96/2=48\), \(a_3=48/2=24\), and \(a_4=24/2=12\). Option 24 is \(a_3\), not \(a_4\). Exam tip: apply the recursive rule step by step from the given first term until the required term is reached.

In the recursive rule, add the current value of n at each step. Thus, a_2=4+1=5, a_3=5+2=7, a_4=7+3=10, and a_5=10+4=14. Therefore, 14 is correct. The value 16 would require adding 6 in the final step instead of 4, which does not follow the given rule. Exam tip: while finding successive terms, use n=1, 2, 3, ... in order.

The recursive rule subtracts the index at each step. Thus, \(a_2=22-1=21\), \(a_3=21-2=19\), and \(a_4=19-3=16\). Therefore, 16 is correct. Note that 19 is \(a_3\), not \(a_4\). Exam tip: to find \(a_4\), apply the rule successively for \(n=1,2,3\).

In the recursive rule, add \(3n\) to the current term at each step. \(a_2=5+3(1)=8\), \(a_3=8+3(2)=14\), and \(a_4=14+3(3)=23\). Therefore, the correct answer is 23. The option 21 would result from not applying the final increment \(3n\) correctly. Exam tip: to find \(a_4\), apply the rule successively for \(n=1,2,3\).

Using n=1, 2, and 3 in the recursive rule gives a_2=8+1=9, a_3=9+3=12, and a_4=12+5=17. Therefore, 17 is correct. The option 16 is incorrect because the successive odd numbers 1, 3, and 5 must be added. Exam tip: To find a_4 from a_1, apply the recursive rule three times.

By the recursive rule, each next term is 4 times the preceding term. Thus, \(a_2=4\times3=12\) and \(a_3=4\times12=48\). Therefore, 48 is correct. The value 24 would result from multiplying \(a_2\) by 2, so it does not follow the given rule. Exam tip: to find \(a_3\), apply the rule twice starting from \(a_1\).

The recursive rule means that each new term is found by multiplying the previous term by 2 and then adding 1. Thus, \(a_2=2\times3+1=7\), and \(a_3=2\times7+1=15\). Therefore, 15 is correct. A value such as 13 can result from applying the rule incorrectly. Exam tip: to find \(a_3\), calculate \(a_2\) first.

Apply the recursive rule step by step: \(a_2=2\times7-3=11\). Then \(a_3=2\times11-3=19\). Therefore, the correct answer is 19. The value 22 would result from forgetting to subtract 3 in the second step. Exam tip: for every new term, first multiply the previous term by 2 and then subtract 3.

The governing concept is a recursive sequence rule. The relation a_{n+1}=a_n+7 says that every new term is obtained by adding 7 to the immediately preceding term. Starting with a_1=2, calculate successively: a_2=2+7=9, a_3=9+7=16, a_4=16+7=23, and a_5=23+7=30. Therefore option B is correct. The same result follows from the arithmetic-sequence formula a_n=a_1+(n−1)d: a_5=2+4(7)=30. Option A is the fourth term, while options C and D add 7 too many times. The recurrence must be applied exactly four times after the first term to reach the fifth term.

The recursive rule makes each new term 6 less than the preceding term. Thus, the terms are 60, 54, 48, 42, 36, 30. Therefore, the sixth term, (a_6), is 30. The value 36 is the fifth term, so it is a close but incorrect option. Exam tip: from the first term to the sixth term, subtract 6 five times.

This is governed by a second-order recursive rule: every term equals the sum of the two immediately preceding terms. The initial values are a_1 = 4 and a_2 = 7. Therefore, a_3 = 7 + 4 = 11. Then a_4 = a_3 + a_2 = 11 + 7 = 18. Finally, a_5 = a_4 + a_3 = 18 + 11 = 29. Hence option D is correct. Option A is only a_4, option B does not arise from the stated sequence, and option C reflects an arithmetic or indexing mistake. It is important to use the two latest available terms at each step, not repeatedly add the original starting values.