Recursive rule

Correct answer: D. 60

Explanation: Using the recursive rule, \(a_2=2\times6+4=16\), \(a_3=2\times16+4=36\), and \(a_4=2\times36+4=76\). Hence, \(a_4-a_2=76-16=60\), so option D is correct. \(56\) results from an incorrect calculation of the terms. Exam tip: List the required terms in order before finding their difference.

Correct answer: C. 20

Explanation: Given \(a_1=6\), the recursive rule gives \(a_2=2\times6+4=16\) and \(a_3=2\times16+4=36\). Therefore, \(a_3-a_2=36-16=20\). Note that 16 is only the second term, not the required difference. Exam tip: in recursive-sequence questions, calculate the required terms in order before finding their difference.

Correct answer: B. 53

Explanation: By the recursive rule, each new term is obtained by adding the previous two terms and 4. Thus, \(a_3=5+9+4=18\), \(a_4=9+18+4=31\), and \(a_5=18+31+4=53\). Therefore, the correct answer is 53. A value such as 49 can result from missing the constant 4 in a step. Exam tip: write each term in order and include the stated constant at every step.

Correct answer: C. 31

Explanation: The recursive rule requires adding the previous two terms and then adding 4 each time. First, \(a_3=9+5+4=18\). Then, \(a_4=18+9+4=31\). Therefore, the correct answer is \(31\). The value \(29\) would result from adding only 2 instead of 4 in the final step, so it does not follow the given rule. Exam tip: write each intermediate term in order until you reach the required term.

Correct answer: A. It is an arithmetic progression with common difference 4.

Explanation: The rule adds 4 to every preceding term, so the difference between consecutive terms is constantly 4. Hence it is an arithmetic progression. A difference of \(-4\) would make terms decrease. Exam tip: identify an AP by checking for a constant difference.

Correct answer: A. \(a_{n+1}=a_n+d\), जहाँ \(d\) नियत है

Explanation: In option A, the difference between consecutive terms is \(a_{n+1}-a_n=d\), which is constant; hence it is an arithmetic progression. In option C, the difference \(n\) changes. Exam tip: check for a constant difference in an AP.

Correct answer: C. 21

Explanation: Apply the recursive rule using the correct value of \(n\) at each step. For \(n=1\), \(a_2=4+7(1)-2=9\). Then, with \(n=2\), \(a_3=9+7(2)-2=21\). Hence, 21 is correct. A value such as 20 can result from using an incorrect value for \(7n-2\) in the second step. Exam tip: use \(n=1\) to find \(a_2\), and \(n=2\) to find \(a_3\).

Correct answer: B. \(127\)

Explanation: Using the recursive rule with \(n=1\), \(a_2=150-(5\times1+4)=141\). Then, with \(n=2\), \(a_3=141-(5\times2+4)=141-14=127\). Therefore, \(127\) is correct. A close distractor such as \(124\) can result from calculating \(5n+4\) incorrectly in the second step. Exam tip: use the next value of \(n\) for each successive term.

Correct answer: A. (7)

Explanation: (a_3=3\times2-2\times1+1=5) and (a_4=3\times5-2\times2+1=12). The correct option should include (12).

Correct answer: A. \(a_n=a_{n-1}+3\), for \(n\geq2\)

Explanation: In an arithmetic progression, each term is obtained by adding a fixed common difference to the previous term. Since the difference is 3, \(a_n=a_{n-1}+3\). Exam tip: look for constant addition in a recursive AP rule.

Correct answer: C. (20)

Explanation: The direct answer is option C, 20. A recursive rule tells us how to move from one term to the next. Starting with \(a_1=5\), we need four additions to reach \(a_5\): use \(n=1,2,3,4\). The additions are \(\frac{3(1)}2=\frac32\), \(\frac{3(2)}2=3\), \(\frac{3(3)}2=\frac92\), and \(\frac{3(4)}2=6\). Their total is \(\frac32+3+\frac92+6=15\), or \(\frac{3(1+2+3+4)}2=15\). Therefore \(a_5=5+15=20\). Option C is correct. Option A, 18, misses 2; option B, 19, misses 1; and option D, 21, adds 1 too much. The common mistake is to use only three increments, but moving from term 1 to term 5 requires four steps. Memory cue: for \(a_5\), add the changes for indices 1 through 4.

Correct answer: B. (25)

Explanation: The direct answer is option B, 25. Begin with \(a_1=40\). The rule subtracts \(\frac{3n}{2}\), and reaching the fifth term requires n=1,2,3,4. The subtractions are \(\frac32,3,\frac92,6\). Their total is \(\frac32+3+\frac92+6=15\), also \(\frac{3(1+2+3+4)}2=15\). Thus \(a_5=40-15=25\). Option B is correct. Option A, 23, subtracts 17 instead of 15; option C, 27, subtracts only 13; and option D, 29, subtracts only 11. Notice that every change is subtracted, because the recurrence contains a minus sign. A frequent mistake is to add the changes or to stop after three changes. Exam cue: from the first term to the fifth term, use exactly four updates, with indices 1 to 4.

Correct answer: C. \(42\)

Explanation: In the recursive rule, use the current index to find the next term. Thus, \(a_2=1\times2+3=5\), \(a_3=2\times5+3=13\), and \(a_4=3\times13+3=42\). Therefore, the correct answer is \(42\). An option such as \(39\) results from using the multiplier 3 or the added 3 incorrectly while finding \(a_4\). Exam tip: for \(a_{n+1}\), substitute \(n=1\), then \(n=2\), and then \(n=3\) in order.

Correct answer: A. \(a_{n+1}=r a_n\)

Explanation: Under \(a_{n+1}=r a_n\), the ratio \(a_{n+1}/a_n=r\) remains constant, so the sequence is geometric. In option B, the difference is constant, which defines an arithmetic progression. Exam tip: check whether a rule fixes a ratio or a difference.

Correct answer: A. \(a_1=3,\ a_{n+1}=2a_n+2\)

Explanation: The correct rule is \(a_1=3,\ a_{n+1}=2a_n+2\). It gives \(2\times3+2=8\), then \(2\times8+2=18\), and \(2\times18+2=38\), matching every listed term. Option B gives \(7\) as the second term, while option C has the wrong first term. Exam tip: test a recursive rule by generating at least three consecutive terms.

Correct answer: B. \(a_1=10,\ a_{n+1}=2a_n-3\)

Explanation: In option B, \(a_1=10\). Applying the rule gives \(2(10)-3=17\), \(2(17)-3=31\), and \(2(31)-3=59\), so it matches every given term. In option A, the next term would be 24, while option D has the wrong first term. Exam tip: when checking a recursive rule, verify the initial term and at least two subsequent terms.

Correct answer: B. 5th term

Explanation: The recursive rule subtracts 1², 2², 3², and so on. Thus, a₁=100, a₂=100−1=99, a₃=99−4=95, a₄=95−9=86, and a₅=86−16=70. Therefore, 70 is the fifth term. The fourth term is 86, so it is a close but incorrect option. Exam tip: in a recursive rule, use the value of n while moving from one term to the next.

Correct answer: C. 40

Explanation: Apply the recursive rule step by step: \(a_3=2+7+3=12\), \(a_4=7+12+4=23\), and \(a_5=12+23+5=40\). Therefore, the correct answer is 40. A value such as 38 can result from forgetting to add 5 while finding \(a_5\). Exam tip: for every new term, add both preceding terms and the index of the term being found.

Correct answer: C. 23

Explanation: First find the third term using the recursive rule: (a_3=7+2+3=12). Then, (a_4=a_3+a_2+4=12+7+4=23). Therefore, the correct answer is 23. The value 21 may result from forgetting to add 4 in the final step, but the rule requires adding the current index n. Exam tip: apply a recursive rule step by step until you reach the required term.

Correct answer: B. \(32\)

Explanation: Given \(a_1=9\), first use \(n=1\) in the recursive rule: \(a_2=a_1+a_1+1^2=9+9+1=19\). Next, use \(n=2\): \(a_3=a_2+a_1+2^2=19+9+4=32\). Hence, the correct answer is \(32\). A choice such as \(34\) can result from an error in using the square term or the previous term. Exam tip: substitute the correct value of \(n\) at every step of a recurrence relation.

Correct answer: B. 19

Explanation: Given \(a_1=9\), substitute \(n=1\) in the recursive rule to find the second term: \(a_2=a_1+a_1+1^2=9+9+1=19\). Hence, 19 is correct. The value 17 would result from incorrectly omitting the \(n^2\) term. Exam tip: To find \(a_{n+1}\), first substitute the appropriate value of \(n\) into the recursive rule.

Correct answer: C. 59

Explanation: The recursive rule generates each new term from the preceding term. Thus, \(a_2=2\times6+1=13\), \(a_3=2\times13+2=28\), and \(a_4=2\times28+3=59\). Hence, the correct answer is 59. An answer such as 57 can result from incorrectly adding 1 instead of using \(n=3\) in the final step. Exam tip: write the value of \(n\) for each transition before substituting it into the rule.

Correct answer: B. 42

Explanation: The governing concept is a recursive sequence: each term is calculated from the preceding term using the current value of n. Begin with a₁ = 80. For n = 1, subtract 3(1)+2 = 5, so a₂ = 80 - 5 = 75. For n = 2, subtract 3(2)+2 = 8, giving a₃ = 75 - 8 = 67. For n = 3, subtract 3(3)+2 = 11, so a₄ = 67 - 11 = 56. Finally, for n = 4, subtract 3(4)+2 = 14, giving a₅ = 56 - 14 = 42. Thus option B is correct. The successive deductions are 5, 8, 11, and 14; each rises by 3. Skipping a transition or using the wrong index can produce the other choices.

Correct answer: A. 61

Explanation: Substituting \(n=1,2,3,4\) into the recursive rule gives successive increments of \(4,10,18,28\). Thus, \(a_2=1+4=5\), \(a_3=5+10=15\), \(a_4=15+18=33\), and \(a_5=33+28=61\). Therefore, 61 is correct. A value such as 63 would result from an incorrect addition at one of the steps. Exam tip: to find \(a_5\), apply the rule from \(n=1\) through \(n=4\).

Correct answer: A. \(a_{n+1}=a_n+7\)

Explanation: In an arithmetic progression, the difference between consecutive terms is constant. The rule \(a_{n+1}=a_n+7\) adds exactly 7 at every step, so its common difference is 7. In \(a_n+n\), the difference changes with \(n\). Exam tip: compute \(a_{n+1}-a_n\); it must be a constant for an AP.