Recursive rule

For n=1, a_2=25-(2×1+1)=22. Then, for n=2, a_3=22-(2×2+1)=22-5=17. Hence, 17 is correct. A value such as 19 can result from incorrectly using a number other than 5 in the second step. Exam tip: increase the value of n by 1 for each successive term.

The second term is 5, and each following term is 2 more than the preceding term. Thus, \(a_3=5+2=7\) and \(a_4=7+2=9\). Therefore, the correct answer is 9. Note that 7 is \(a_3\), not \(a_4\). Exam tip: In a recursive rule, apply the stated operation step by step from the given term.

By the recursive rule, each new term is twice the preceding term. Thus the sequence is \(1, 2, 4, 8, 16\), so \(a_5=16\). The value \(8\) is the fourth term, \(a_4\), not the fifth term. Exam tip: write the term numbers while extending a recursively defined sequence.

A recursive rule defines a term by using earlier terms. Here, every term after the first two is found by adding the two immediately preceding terms. The starting values are already given, so the sequence can be built one position at a time. Keeping the position labels visible helps prevent skipping a term or confusing the required term with the previous one.

Start with \(a_1=3\) and \(a_2=5\). Then \(a_3=3+5=8\), \(a_4=5+8=13\), \(a_5=8+13=21\), and \(a_6=13+21=34\). Therefore the correct choice is D, \(34\). The other numerical choices result from stopping too early or making an addition error.

The governing concept is a recursive rule, in which each term is obtained from the preceding term. Examine the consecutive changes in the sequence: 7 − 4 = 3, 10 − 7 = 3, and 13 − 10 = 3. Since the same difference occurs each time, the recursive rule is “add 3 to the previous term.” Applying this rule to the last known term gives 13 + 3 = 16. Therefore, option C is correct. This sequence is also an arithmetic progression with common difference 3, although the question specifically highlights the step-by-step recursive construction. Option A, 14, changes by only 1; option B, 15, changes by 2; and option D, 17, changes by 4, so none preserves the established rule.

The governing pattern is a recursive operation applied to the previous term. Each listed term is one-half of the preceding term: 16 = 32 ÷ 2, 8 = 16 ÷ 2, and 4 = 8 ÷ 2. Continuing exactly the same rule, the next term is 4 ÷ 2 = 2. Therefore option B is correct. Option D merely repeats the last known term, while option C does not maintain the constant ratio. Option A would be obtained only after applying the division rule twice more: 4 becomes 2 and then 1. The sequence is also geometric with common ratio 1/2, but the essential task is to continue its recursive rule.

Check the difference between consecutive terms: \(12-6=6\), \(18-12=6\), and \(24-18=6\). Thus, each new term is obtained by adding \(6\) to the previous term, so the recursive rule is \(a_{n+1}=a_n+6\). The number \(12\) is the second term, not the amount added. Exam tip: to find the number added in a recursive rule, subtract one consecutive term from the next.

Check the differences between consecutive terms: \(15-18=-3\), \(12-15=-3\), and \(9-12=-3\). Therefore, 3 is subtracted from each previous term, so the recursive rule is \(a_{n+1}=a_n-3\). If 2 were subtracted, the next term would be 16, which does not match the sequence. Exam tip: find the difference between two consecutive terms to identify a recursive rule.

Each term is obtained by multiplying the previous term by 3: \(3\times3=9\), \(9\times3=27\), and \(27\times3=81\). Thus, the recursive rule is \(a_n=3a_{n-1}\), so the multiplier is 3. Although 9 is a term of the sequence, it is not the multiplier. Exam tip: divide a term by the preceding term, for example \(27\div9=3\), to find the multiplier.

By the recursive rule, each new term is obtained by adding 7 to the preceding term. Thus, \(a_2=11+7=18\) and \(a_3=18+7=25\). Therefore, the correct answer is 25. Option 18 is the second term, so it is a close but incorrect distractor. Exam tip: to find \(a_3\) from \(a_1\), apply the rule twice.

The recursive rule adds 9 to each preceding term. Thus, a_2=5+9=14, a_3=14+9=23, and a_4=23+9=32. Therefore, 32 is correct. Note that 23 is the third term, not the fourth. Exam tip: from a_1 to a_4, apply the rule three times.

The terms are (90,30,10,\frac{10}{3}), so (a_4=\frac{10}{3}). None of the listed options is correct.

To evaluate a recursively defined sequence, apply the stated operation once for each transition between the starting index and the requested index. The sequence begins with a₁ = 90. Applying the rule once gives a₂ = 90 ÷ 3 = 30. Applying it a second time gives a₃ = 30 ÷ 3 = 10. Therefore option A is correct. Option C is the value of a₂, so it results from stopping one step too early. Option B does not arise from division by 3. Option D would be obtained by dividing a₃ by 3 once more, so it corresponds to a₄ rather than a₃. Correct indexing is essential in recurrence questions.

Apply the recursive rule successively for n=1, 2, and 3. Thus, a_2=2+1+1=4, a_3=4+2+1=7, and a_4=7+3+1=11. Therefore, the correct answer is 11. The value 10 may result from incorrectly treating the added quantity as fixed at every step. Exam tip: To find a_4, start from a_1 and apply the rule three times.

In the recursive rule, first put \(n=1\): \(a_2=a_1+2(1)+2=3+4=7\). Then put \(n=2\): \(a_3=a_2+2(2)+2=7+6=13\). Therefore, the correct answer is 13. The value 11 can result from not updating the value of \(n\) correctly in the second step. Exam tip: use the next integer value of \(n\) when finding each new term.

The governing concept is a second-order recursive sequence. Every term from the third onward equals the sum of the two terms immediately before it. The initial values are a₁ = 4 and a₂ = 6. Calculate in order: a₃ = a₂ + a₁ = 6 + 4 = 10; a₄ = a₃ + a₂ = 10 + 6 = 16; and a₅ = a₄ + a₃ = 16 + 10 = 26. Hence option C is correct. Option A is the fourth term, so it results from stopping one step too early. Options B and D cannot be obtained by adding the correct pair of preceding terms. The order matters: for a₅, the required pair is a₄ and a₃, not just one earlier term or the original starting values.

The recursive rule adds \(2\) to each preceding term. Thus the sequence is \(4, 6, 8, 10, 12\), so \(a_5=12\). Note that \(10\) is the fourth term, \(a_4\), not the fifth. Exam tip: count terms starting from \(a_1\).

The recursive rule subtracts 3 from each preceding term. Thus, \(a_2=18-3=15\), \(a_3=15-3=12\), and \(a_4=12-3=9\). Therefore, the correct answer is 9. The close distractor 12 is \(a_3\), not \(a_4\). Exam tip: write successive terms in order and check the term number carefully.

Each new term is twice the preceding term. Thus, \(a_2=2\times5=10\) and \(a_3=2\times10=20\). Therefore, 20 is correct. A value such as 15 may result from incorrectly adding instead of multiplying; the rule requires multiplication at every step. Exam tip: write the terms in order and apply the recursive rule one step at a time.

By the recursive rule, each new term is half of the preceding term: \(a_2=24\), \(a_3=12\), \(a_4=6\), and \(a_5=3\). Therefore, the correct answer is 3. The value 6 is \(a_4\), so it is a close but incorrect option. Exam tip: start from \(a_1\) and count the term positions carefully.