Recursive rule

Putting \(n=1\), \(a_2=80-\frac{1\cdot2}{2}-1=80-2=78\). Then, for \(n=2\), \(a_3=78-\frac{2\cdot3}{2}-1=78-4=74\). Hence, the correct answer is 74. The value 76 can result from calculating the first decrement incorrectly. Exam tip: in a recursive rule, substitute the correct value of \(n\) for each successive term.

Using the recursive rule with \(n=1\), \(a_2=\frac{14}{2}+2(1)=9\). Next, with \(n=2\), \(a_3=\frac{9}{2}+2(2)=\frac{17}{2}\). Hence, option C is correct. The value 9 is \(a_2\), not \(a_3\). Exam tip: Increase the value of \(n\) step by step while finding successive terms.

To find \(a_2\), substitute \(n=1\) and \(a_1=16\) into the recursive rule: \(a_2=\frac{a_1}{2}+2(1)=\frac{16}{2}+2=8+2=10\). Therefore, 10 is correct. Option 8 only halves \(a_1\) and misses the \(2n\) term. Exam tip: for \(a_2\), use \(n=1\) in the recurrence.

Here, a_1=4 is fixed, so 3a_1=12. Putting n=1 gives a_2=4+12+1=17. Then, putting n=2 gives a_3=17+12+2=31. Therefore, the correct answer is 31. The value 30 would result from incorrectly using n=1 again in the second step. Exam tip: Use the next value of n for each new term, but keep a_1 fixed.

Substitute n=3 in the recursive rule: \(a_3=a_2+a_1+2(3)\). Therefore, \(a_3=4+3+6=13\). The value 15 can result from using an incorrect value for \(2n\). Exam tip: first substitute the index of the required term, then use the given initial terms carefully.

First, for n=3, a_3=a_2+a_1+2(3)=4+3+6=13. Then, for n=4, a_4=a_3+a_2+2(4)=13+4+8=25. Therefore, the correct answer is 25. A value such as 23 can result from using an incorrect value of n in the term 2n. Exam tip: write the current value of n for each new term in a recursive rule.

Use the recursive rule first to find a_2: \(a_2=4^2-2(4)=16-8=8\). Then, \(a_3=8^2-2(8)=64-16=48\). Therefore, the correct answer is 48. A value such as 40 can result from an arithmetic error. Exam tip: calculate each term first and substitute it carefully into the next step.

In a recursive rule, a new term is obtained from one or more earlier terms, for example \(a_{n+1}=a_n+3\). Option A describes an explicit rule, where a term is given directly in terms of \(n\). Exam tip: look for a previous-term relation.

Put n=1 in the recursive rule: a_2=a_1^2+a_1+1. Therefore, a_2=3^2+3+1=9+3+1=13, so option B is correct. The value 11 would result from forgetting to add 1. Exam tip: to find a_2, substitute n=1 in the rule and then use the given value of a_1.

The governing concept is recursive evaluation of a sequence. Since 4 is added whenever we move from one term to the next, the sequence is also an arithmetic progression with first term a₁ = 6 and common difference d = 4. To reach a₉ from a₁, the rule must be applied 9 − 1 = 8 times, not nine times. Therefore, a₉ = 6 + 8(4) = 6 + 32 = 38. Listing the terms gives 6, 10, 14, 18, 22, 26, 30, 34, 38, which confirms the calculation. Thus option C is correct. Option A is a nearby earlier term, while option D can result from counting one extra step.

The recurrence adds the constant 8 at every step, so it forms an arithmetic progression whose first term is 15 and common difference is 8. Its general term is aₙ = 15 + (n − 1)8. To find the position of 71, set 15 + 8(n − 1) = 71. Subtracting 15 gives 8(n − 1) = 56; dividing by 8 gives n − 1 = 7, so n = 8. Hence 71 is the eighth term and option B is correct. Direct checking gives the sequence 15, 23, 31, 39, 47, 55, 63, 71. The first term is counted as position 1, not as one application of the addition rule, so answers 9th and 10th involve over-counting.

Using the recursive rule: \(a_2=2(5)+3=13\), \(a_3=2(13)+3=29\), and \(a_4=2(29)+3=61\). Therefore, \(a_4-a_2=61-13=48\), so option D is correct. Option 42 is not the correct difference between \(a_4\) and \(a_2\). Exam tip: List the required terms in order before finding the requested difference.

Given \(a_1=5\) and \(a_{n+1}=2a_n+3\), we get \(a_2=2\times5+3=13\) and \(a_3=2\times13+3=29\). Therefore, \(a_3-a_2=29-13=16\). A value such as 14 can result from an error while calculating the second term. Exam tip: in a recursive sequence, find each required term step by step from the preceding term.

Using the recursive rule, find the preceding terms first: \(a_3=7+4+3=14\), \(a_4=14+7+3=24\), and \(a_5=24+14+3=41\). Therefore, the correct answer is 41. A value such as 43 results from incorrectly adding one of the previous terms or the constant 3. Exam tip: use exactly the two immediately preceding terms at every step.

Use the recursive rule step by step. First, \(a_3=7+4+3=14\). Then, \(a_4=14+7+3=24\). Therefore, the correct answer is 24. The value 21 results from omitting the \(+3\) in the recurrence. Exam tip: write each required term in order before finding the final term.

In option A, \(a_{n+1}-a_n=7\) is constant, so the sequence is an AP with common difference 7. In option C, the difference changes with \(n\). Exam tip: check whether consecutive differences are fixed.

In the recursive rule, add the stated value for each step to the current term. Thus, \(a_2=3+6(1)-1=8\), \(a_3=8+6(2)-1=19\), and \(a_4=19+6(3)-1=36\). Therefore, 36 is correct. Options such as 35 or 37 can result from miscalculating the final increment, \(6\times3-1=17\). Exam tip: substitute the correct value of \(n\) before finding \(a_{n+1}\).

Using the recursive rule with n=1, a_2=a_1+6(1)-1=3+5=8. Then, with n=2, a_3=a_2+6(2)-1=8+11=19. Therefore, the correct answer is 19. The value 17 may result from incorrectly using an increment of 9 in the second step. Exam tip: To find a_3, apply the rule successively for n=1 and n=2.

Using n=1, 2, and 3 in the recursive rule, the amounts subtracted are 7, 11, and 15 respectively. Thus, a_2=150-7=143, a_3=143-11=132, and a_4=132-15=117. Therefore, the correct answer is 117. A value such as 114 can result from using an incorrect subtraction in one of the steps. Exam tip: use the next value of n for each new term.

Using the recursive rule with \(n=1\), \(a_2=150-(4\times1+3)=143\). Then, with \(n=2\), \(a_3=143-(4\times2+3)=143-11=132\). Therefore, 132 is correct. A value such as 129 can result from using an incorrect amount in the second subtraction. Exam tip: increase the value of \(n\) step by step while finding successive terms.