Recursive rule

The recursive rule makes each new term 20 less than the previous term. Thus, \(a_1=80\), \(a_2=60\), \(a_3=40\), and \(a_4=20\). Therefore, the correct answer is 20. Note that 40 is \(a_3\), not \(a_4\). Exam tip: start from \(a_1\) and apply the rule once for each next term required.

By the recursive rule, each new term is 3 greater than the previous term. Thus, \(a_2=10\) and \(a_5=19\). Therefore, \(a_5-a_2=19-10=9\). The value 12 would correspond to a gap of four steps, but there are only three steps from \(a_2\) to \(a_5\). Exam tip: in an arithmetic progression, the difference between two terms equals the number of steps times the common difference.

By the recursive rule, each new term is 4 more than the previous term. Thus the terms are 5, 9, 13, and 17. Therefore, (a_3+a_4=13+17=30), so 30 is correct. The value 28 would result from incorrectly taking the fourth term as 15 instead of 17. Exam tip: Write the sequence up to the required terms before finding their sum.

The recursive rule adds 4 to each preceding term. Thus, \(a_2=5+4=9\), \(a_3=13\), and \(a_4=17\). Therefore, \(a_2+a_4=9+17=26\). Option 24 may result from not applying the increase of 4 correctly. Exam tip: first write the required terms in order, then add them.

Putting \(n=1\) in the recursive rule gives \(a_2=a_1^2\). Therefore, \(a_2=3^2=9\), so option B is correct. \(6\) is twice 3, but the rule requires squaring the previous term. Exam tip: to find \(a_2\), substitute \(n=1\) directly into the rule.

Putting \(n=1\) in the recursive rule gives \(a_2=a_1^2+2\). Therefore, \(a_2=2^2+2=4+2=6\). Option 4 is only the square of 2; the required addition of 2 has not been made. Exam tip: to find \(a_2\), substitute the initial term \(a_1\) into the recursive rule.

The recursive rule subtracts 25 from each preceding term. Thus, \(a_2=120-25=95\) and \(a_3=95-25=70\). Therefore, 70 is correct. The value 75 would result from subtracting 20 in the second step, so it is not correct. Exam tip: to find \(a_3\) from \(a_1\), apply the recursive rule twice.

The recursive rule adds 8 to each preceding term. Thus, \(a_2=13+8=21\), \(a_3=21+8=29\), and \(a_4=29+8=37\). Therefore, the correct answer is \(37\). Option \(29\) is a close distractor because it is \(a_3\), not \(a_4\). Exam tip: from \(a_1\) to \(a_4\), apply the rule three times.

Each new term is 5 times the previous term. Thus, \(a_2=5\times3=15\) and \(a_3=5\times15=75\). Therefore, 75 is correct. Although 45 may seem plausible, it is not obtained from the given recursive rule. Exam tip: to find \(a_3\) from \(a_1\), apply the rule twice.

Option A gives an initial term and forms every next term by adding 4 to the previous term, so it is recursive. Option B is an explicit formula. Exam tip: look for a reference to the previous term in a recursive rule.

Put \(n=1\) in the recursive rule: \(a_2=\frac{a_1}{2}+2\). Therefore, \(a_2=\frac{36}{2}+2=18+2=20\). Option 18 is only half of 36; the required addition of 2 has been missed. Exam tip: substitute the given previous term directly into the recurrence rule.

In the recursive rule, use the current value of n at each step. Thus, a_2=2+4(1)=6, a_3=6+4(2)=14, and a_4=14+4(3)=26. Therefore, the correct answer is 26. A value such as 22 can result from not applying 4n with the correct n at every step. Exam tip: To find a_4, apply the rule successively for n=1, 2, and 3.

Apply the recursive rule with n=1 first: a_2=a_1-(1+2)=30-3=27. Next, with n=2, a_3=a_2-(2+2)=27-4=23. Therefore, 23 is the correct answer. Note that 27 is only the second term, a_2, not the third term. Exam tip: increase the value of n by 1 at every recursive step.

The recursive rule adds \(3\) to each preceding term. Thus the sequence is \(5, 8, 11, 14, 17\), so \(a_5=17\). The value \(14\) is the fourth term, \(a_4\), making it a close but incorrect option. Exam tip: For a recursive sequence, list the terms one by one until the required term is reached.

By the recursive rule, each new term is three times the previous term. Thus, \(a_3=3\times4=12\) and \(a_4=3\times12=36\). Therefore, 36 is correct. The value 24 is not obtained from the stated rule; 12 is \(a_3\), not \(a_4\). Exam tip: in recursive-rule questions, write each successive step before finding the required term.

By the recursive rule, each new term is the sum of the two preceding terms. The sequence is 5, 7, 12, 19, 31, 50. Therefore, \(a_6=31+19=50\). Option 31 is \(a_5\), not \(a_6\). Exam tip: Write each term in order and count the term positions carefully.

The governing concept is a recursive rule: each new term is calculated from earlier terms, rather than from n directly. Start with a_1 = 5 and a_2 = 7. Then a_3 = a_2 + a_1 = 7 + 5 = 12. Next, a_4 = a_3 + a_2 = 12 + 7 = 19. Finally, a_5 = a_4 + a_3 = 19 + 12 = 31. Therefore, option C is correct. Option A may come from stopping too early or using an incorrect addition, option B is the preceding-term value in another possible calculation, and option D does not follow the stated recurrence. The order of the two previous terms does not change their sum, but both must be included.

The governing concept is a constant-difference recursive sequence. Each term increases by 4: 9 + 4 = 13, 13 + 4 = 17, and 17 + 4 = 21. Continuing the same rule gives 21 + 4 = 25. Therefore option C is correct. The values 23, 24, and 26 would require changes of 2, 3, or 5 rather than the established constant increase of 4.

In this sequence, each term is half of the previous term: 64 to 32, 32 to 16, and 16 to 8. Therefore, the next term is \(8 \div 2=4\). Option 6 is not correct because it is not half of 8. Exam tip: in a recursive rule, identify the operation from one term to the next.

Check the difference between consecutive terms: 16−8=8, 24−16=8, and 32−24=8. Thus, each new term is obtained by adding 8 to the previous term, so the recursive rule is \(a_{n+1}=a_n+8\). Adding 16 would give 32 after 16 instead of 24, so it is not correct. Exam tip: find the difference between two consecutive terms to identify the added number in a recursive rule.