Recursive rule

Correct answer: C. 126

Explanation: In a recursive rule, each new term is calculated from the preceding term. \(a_2=3(4)+1=13\), \(a_3=3(13)+2=41\), and \(a_4=3(41)+3=126\). Therefore, the correct answer is 126. A value such as 122 can result from forgetting to add \(+3\) in the final step. Exam tip: at each step, use the value of \(n\) from the subscript of the preceding term.

Correct answer: B. (100)

Explanation: The rule subtracts a different amount at each step. Since the first term is \(a_1=150\), use \(n=1,2,3,4\) to reach \(a_5\). The subtracted quantities are found from \(n^2+2n\): for these four indices they are 3, 8, 15, and 24. Writing them separately reduces the chance of using the wrong index or missing a subtraction.

Apply the rule successively: \(a_2=150-3=147\), \(a_3=147-8=139\), \(a_4=139-15=124\), and \(a_5=124-24=100\). The same result comes from \(150-(3+8+15+24)=150-50=100\). Therefore option B is correct. The answer is not obtained by subtracting the same amount each time.

Correct answer: B. 142

Explanation: This is a second-order recursive sequence because each new term depends on the two immediately preceding terms. The recurrence is aₙ = 2aₙ₋₁ + 3aₙ₋₂, so the previous term is multiplied by 2 and the term before it by 3. Starting with a₁ = 2 and a₂ = 5, calculate a₃ = 2(5) + 3(2) = 16. Next, a₄ = 2(16) + 3(5) = 32 + 15 = 47. Finally, a₅ = 2(47) + 3(16) = 94 + 48 = 142. Hence option B is correct. The other choices can result from omitting one preceding term, interchanging coefficients, or making an arithmetic error. Both initial values are essential because this recurrence has order two.

Correct answer: C. 34

Explanation: First, substitute \(n=3\) in the recurrence: \(a_3=3\times3-1+3=11\). Then, for \(n=4\), \(a_4=3\times11-3+4=34\). Therefore, the correct answer is 34. The distractor 33 results from omitting the final \(+4\). Exam tip: write the current value of \(n\) separately at each step before substituting.

Correct answer: C. 66

Explanation: Using n=1, 2, 3 and 4 in the recursive rule, the added values are 3, 8, 17 and 32 respectively. Thus, a_2=9, a_3=17, a_4=34, and a_5=34+32=66. Therefore, the correct answer is 66. The value 69 can result from an incorrect addition in the final step. Exam tip: calculate 2^n and n^2 separately at each step before adding them.

Correct answer: A. 48

Explanation: Substitute n=1, 2, 3, and 4 in the recursive rule. The added values are \(1!+1=2\), \(2!+2=4\), \(3!+3=9\), and \(4!+4=28\). Hence, \(a_2=7\), \(a_3=11\), \(a_4=20\), and \(a_5=20+28=48\). Therefore, 48 is correct. A value such as 50 results from an incorrect addition in one of the steps. Exam tip: to find \(a_5\), apply the rule only for \(n=1\) through \(4\).

Correct answer: C. 37

Explanation: Substituting n=1, 2, 3, and 4 into the recursive rule gives the subtracted values 3, 6, 12, and 32. Thus, a_2=90-3=87, a_3=87-6=81, a_4=81-12=69, and a_5=69-32=37. Therefore, 37 is correct. The value 34 can result from an error in the final subtraction; for n=4, 4!=24, not 6. Exam tip: to find a_5, apply the rule only from n=1 through n=4.

Correct answer: B. 37

Explanation: Using n=1 in the recursive rule, a_2=4a_1-(1+1)=4×3-2=10. Then, using n=2, a_3=4a_2-(2+1)=4×10-3=37. Hence, 37 is the correct option. The value 39 would result from incorrectly subtracting 1 instead of 3 in the second step. Exam tip: write the current value of n before substituting it at each step.

Correct answer: C. 47

Explanation: First, put n=3: a_3=6+2(2)+3^2=19. Then, for n=4, a_4=19+2(6)+4^2=19+12+16=47. Therefore, 47 is correct. Using only the previous term gives an incorrect result because the rule also includes twice a_{n-2} and n^2. Exam tip: substitute the correct value of n at each step and write every term separately.

Correct answer: C. 48

Explanation: Apply the recursive rule successively for n=1, 2, and 3. We get a_2=8+1(1+2)+2^1=13, a_3=13+2(2+2)+2^2=25, and a_4=25+3(3+2)+2^3=48. Hence, 48 is correct. A value such as 45 may result from incorrectly evaluating 2^3 in the final step. Exam tip: to find a_4 from a_1, apply the rule exactly three times.

Correct answer: C. (42)

Explanation: The recurrence has two parts: double the previous term, then add 3n − 2. To find a₃, use the rule first with n = 1 and then with n = 2. The value of n belongs to the step being performed, while the sequence term on the right is the immediately preceding term.

Starting from a₁ = 9, for n = 1 we get a₂ = 2(9) + 3(1) − 2 = 18 + 1 = 19. For n = 2, use a₂ = 19: a₃ = 2(19) + 3(2) − 2 = 38 + 6 − 2 = 42. Thus the terms needed are 9, 19, and 42, so option C is correct. A common error is to add 3n − 2 to the original term again instead of using the newly calculated a₂.

Correct answer: A. 12

Explanation: Use the recursive rule first with \(n=1\) to find \(a_2\): \(a_2=\frac{30}{3}+2(1)^2=10+2=12\). Then use \(n=2\): \(a_3=\frac{12}{3}+2(2)^2=4+8=12\). Therefore, the correct answer is 12. A value such as 14 can result from evaluating the \(2n^2\) term incorrectly. Exam tip: substitute the correct value of \(n\) at every step of a recursive sequence.

Correct answer: B. 40

Explanation: First, put n=1: a_2=5+3(5)+1^2=21. Then put n=2: a_3=21+3(5)+2^2=40. Therefore, the correct answer is 40. The term 3a_1 remains 15 at every step; only n^2 changes. Exam tip: In a recursive rule, calculate each preceding term step by step until the required term is reached.

Correct answer: A. 63

Explanation: First substitute \(n=3\) in the recursive rule: \(a_3=2(5)+2+3^2=21\). Then, for \(n=4\), \(a_4=2(21)+5+4^2=42+5+16=63\). A value such as 66 can result from an error while calculating \(a_3\). Exam tip: always add the square of the current value of \(n\) separately.

Correct answer: B. 39

Explanation: Apply the recursive rule first to a_1=3: a_2=3^2-2(3)+4=9-6+4=7. Then use a_2=7: a_3=7^2-2(7)+4=49-14+4=39. Therefore, the correct answer is 39. A value such as 42 results from an arithmetic error. Exam tip: to find a_3, apply the rule successively to a_1 and then to a_2.

Correct answer: D. 14

Explanation: Putting \(n=1\) in the recursive rule gives \(a_2=a_1^2-a_1+2\). Therefore, \(a_2=4^2-4+2=16-4+2=14\). Option 12 can result from an error in evaluating \(4^2\) or in the final addition. Exam tip: to find \(a_2\), substitute the value of \(a_1\) into the rule.

Correct answer: A. (12)th

Explanation: (79-13=66), and (66\div6=11) steps, so the term number is (12). Exam tip: add (1) to the number of steps.

Correct answer: B. (107)

Explanation: For (a_{12}), (9) is added (11) times, so (a_{12}=107). Exam tip: the rule is applied one less time than the term number.

Correct answer: D. 78

Explanation: Use the recursive rule to find the required terms: \(a_2=2(4)+9=17\), \(a_3=2(17)+9=43\), and \(a_4=2(43)+9=95\). Therefore, \(a_4-a_2=95-17=78\), so option D is correct. A common error is to stop at \(a_3\) instead of calculating the required fourth term. Exam tip: in a recursive sequence, generate terms step by step until both requested terms are obtained.

Correct answer: A. 61

Explanation: The recursive rule adds the previous two terms and 7. Thus, \(a_3=3+9+7=19\), \(a_4=9+19+7=35\), and \(a_5=19+35+7=61\). Therefore, the correct answer is 61. The option 54 is only \(19+35\), so it misses the required \(+7\). Exam tip: at every step, include both previous terms and the constant 7.

Correct answer: B. 66

Explanation: In a recursive rule, each new term is found from the preceding term. First, \(a_2=7\times2-4=10\). Then, \(a_3=7\times10-4=66\). Therefore, the correct answer is 66. A value such as 62 can result from an incorrect multiplication or subtraction in the final step. Exam tip: always write \(a_2\) before finding \(a_3\).

Correct answer: D. 48

Explanation: The recursive rule adds \(9n-4\) to the current term at each step. For \(n=1\), \(a_2=6+5=11\); for \(n=2\), \(a_3=11+14=25\); and for \(n=3\), \(a_4=25+23=48\). Hence, 48 is correct. A choice such as 45 may result from not using the successive values of \(n\) correctly. Exam tip: to find \(a_4\), apply the rule successively with \(n=1,2,3\).

Correct answer: A. 150

Explanation: Apply the recursive rule successively for \(n=1,2,3\). The amounts subtracted are \(13,20,27\), so \(a_2=210-13=197\), \(a_3=197-20=177\), and \(a_4=177-27=150\). Hence, option A is correct. A nearby value such as \(153\) can result from calculating \(7n+6\) incorrectly in one step. Exam tip: to find \(a_4\) from \(a_1\), apply the rule three times.

Correct answer: C. 41

Explanation: First find the third term using the recursive rule: \(a_3=5(3)-4(2)+3=10\). Then, \(a_4=5(10)-4(3)+3=50-12+3=41\). Therefore, 41 is correct. The nearby distractor 44 can result from not subtracting the \(4a_{n-2}\) term correctly. Exam tip: substitute both previous terms carefully before simplifying.

Correct answer: B. (42)

Explanation: The direct answer is option B, 42. From \(a_1=7\) to \(a_5\), apply the recurrence for n=1,2,3,4. The total increase is \(\frac72+7+\frac{21}2+14=\frac{7(1+2+3+4)}2=35\). Hence \(a_5=7+35=42\). Option B is correct. Option A, 39, is three too small; option C, 45, is three too large; and option D, 48, is six too large. The starting value must be included after calculating the added amount; the total increase alone is 35, not the answer. Also, four increments are needed, not three. Memory cue: sum the coefficients for indices 1 through 4, multiply by \(7/2\), then add the initial 7.