Recursive rule

The governing concept is a recursive sequence in which every term depends on the two immediately preceding terms. Its rule is a_n=a_(n−1)+a_(n−2), with a_1=8 and a_2=13. Apply the rule one step at a time: a_3=8+13=21, a_4=13+21=34, a_5=21+34=55, and a_6=34+55=89. Thus option C is correct. The calculation must be continued successively; adding the first two values only once does not produce the sixth term. Option A is not obtained at any required step, and options B and D do not equal the sum of the fifth and fourth terms. The initial values and recurrence together determine a unique sequence.

The governing concept is a recursive rule: each term is generated from the two immediately preceding terms rather than from a single fixed difference or ratio. The given terms are \(a_1=2,a_2=3,a_3=5,a_4=8,a_5=13\). Continue carefully: \(a_6=a_4+a_5=8+13=21\), \(a_7=a_5+a_6=13+21=34\), and \(a_8=a_6+a_7=21+34=55\). Therefore option C is correct. Option A is the sixth term, and option B is the seventh term, so both result from stopping too early. Option D is the ninth term, obtained after one more addition, which shows why the term number must be tracked at every step.

The governing concept is a recursive addition rule. Starting with the fifth term 11, the next added number is 5, so the sixth term is \(11+5=16\). Add 6 to obtain the seventh term: \(16+6=22\). Add 7 to obtain the eighth term: \(22+7=29\). Therefore option D is correct. A useful equivalent check is \(a_8=1+(1+2+3+4+5+6+7)=1+28=29\). Option B is the seventh term, so it stops one step early. Options A and C do not result from applying the successive additions in order and therefore do not satisfy the stated recursive rule.

The governing idea is to inspect the successive differences rather than assume that the sequence has a constant common difference. The listed differences are -3, -5, -7, and -9; each new difference is 2 smaller than the preceding one. Therefore, the next difference must be -11. The last known term is -9, so the next term is obtained by adding this next difference: -9 + (-11) = -20. Hence option D is correct. Option C would result from using -9 again or making an arithmetic mistake, while the positive options ignore that the sequence is continuing downward. The pattern of odd negative differences confirms the result.

The governing concept is a recursive sequence: each term is calculated from the term immediately before it. The rule is next term = 2 × previous term + 1. Using the last given term, 2 × 127 + 1 = 254 + 1 = 255, so option C is correct. Options A and B result from incorrect arithmetic, while D does not follow the stated rule.

The governing concept is a recursive rule, where the next value depends on the preceding value. Here, next term = 2 × previous term − 1. Applying it to the final term 33 gives 2 × 33 − 1 = 66 − 1 = 65. Therefore option C is correct. Option A would subtract 3, option B omits the subtraction, and option D uses neither the correct operation nor result.

By the recursive rule, each new term is found by adding 3 to the previous term. Thus, \(a_2=2+3=5\), \(a_3=5+3=8\), and \(a_4=8+3=11\). Therefore, 11 is correct. The value 8 is the third term, not the fourth. Exam tip: in recursive-rule questions, list the terms successively until the required term.

The recursive rule subtracts 2 from each preceding term: \(a_1=10\), \(a_2=8\), \(a_3=6\), \(a_4=4\), and \(a_5=2\). Therefore, the correct answer is \(2\). \(4\) is the fourth term, not the fifth. Exam tip: write the term numbers while expanding a recursive sequence.

By the recursive rule, each new term is twice the preceding term. Thus, \(a_2=2\times3=6\), \(a_3=2\times6=12\), and \(a_4=2\times12=24\). Therefore, 24 is correct. Note that 12 is the value of \(a_3\), not \(a_4\). Exam tip: write the terms one by one until you reach the required term.

By the recursive rule, each new term is half of the preceding term. Thus, \(a_2=64/2=32\), \(a_3=32/2=16\), and \(a_4=16/2=8\). Therefore, the correct answer is 8. Option 16 is the third term, \(a_3\), not the fourth term. Exam tip: Start counting from \(a_1\) and apply the given rule at every step.

Each new term is obtained by adding the current index n to the preceding term. Thus, a_2=1+1=2, a_3=2+2=4, a_4=4+3=7, and a_5=7+4=11. Therefore, the correct answer is 11. The value 9 can result from using an incorrect index at a step. Exam tip: when finding a_{n+1}, use the current value of n, not n+1.

In the recursive rule, subtract the current value of n at each step. Thus, \(a_2=20-1=19\), \(a_3=19-2=17\), and \(a_4=17-3=14\). Therefore, 14 is correct. The value 16 would result from stopping after only two subtraction steps. Exam tip: to find \(a_4\), apply the rule successively for \(n=1,2,3\).

Apply the recursive rule successively with n=1, 2, and 3: a_2=4+2(1)=6, a_3=6+2(2)=10, and a_4=10+2(3)=16. Therefore, the correct answer is 16. The value 14 would result from incorrectly using n=2 in the final step; to find a_4, the last step uses n=3. Exam tip: starting from a_1, apply the rule three times to obtain a_4.

Apply the recursive rule successively for n=1, 2, and 3. We get a_2=5+(2×1−1)=6, a_3=6+(2×2−1)=9, and a_4=9+(2×3−1)=14. Therefore, the correct answer is 14. The value 16 would result from incorrectly using n=4 for the next step, which is not needed here. Exam tip: To find a_4 from a_1, apply the rule three times.

Each new term is 3 times the preceding term. Thus, \(a_2=3\times2=6\) and \(a_3=3\times6=18\). Therefore, the correct answer is 18. Option 6 is the second term, not the third. Exam tip: evaluate recursive sequences one term at a time.

The recursive rule says that each new term is twice the previous term plus 1. Thus, a_2=2(1)+1=3, a_3=2(3)+1=7, and a_4=2(7)+1=15. Therefore, the correct answer is 15. The value 7 is a close distractor because it is a_3, not a_4. Exam tip: start with the given first term and apply the rule one step at a time.

Given \(a_1=5\) and \(a_{n+1}=2a_n-1\), we get \(a_2=2\times5-1=9\) and then \(a_3=2\times9-1=17\). Therefore, 17 is correct. A value such as 19 results from using an incorrect previous term. Exam tip: in a recursive rule, calculate each term step by step from the preceding term.

The recursive rule adds 5 to each preceding term. Thus the terms are 2, 7, 12, 17, 22, 27, so the sixth term is 27. The value 22 is the fifth term, making it a close but incorrect distractor. Exam tip: for a constant-add recursive rule, keep adding the same number until the required term.

The recursive rule makes each new term \(4\) less than the preceding term. Thus the terms are \(30, 26, 22, 18, 14\). Therefore, \(a_5=14\). Note that \(16\) is the fourth term, \(a_4\), not the fifth. Exam tip: list the terms in order and count their indices carefully.

By the recursive rule, each new term is the sum of the two preceding terms. Thus, \(a_3=2+3=5\), \(a_4=3+5=8\), and \(a_5=5+8=13\). Therefore, the correct answer is \(13\). Note that \(8\) is \(a_4\), not \(a_5\). Exam tip: write each intermediate term in order until you reach the required term.