Recursive rule

Correct answer: A. \(a_{n+1}=a_n+4\)

Explanation: In an arithmetic progression, each new term is obtained by adding a fixed constant to the previous term. Here, 4 is added every time, so the common difference is constant. Option B multiplies terms by 4, which indicates a geometric progression. Exam tip: look for “+ constant” to identify an AP.

Correct answer: B. 26

Explanation: The governing concept is repeated application of a recurrence whose increment changes with n. To reach a₅ from a₁, use n = 1, 2, 3, and 4. First, a₂ = 2 + 1(2)/2 + 1 = 2 + 1 + 1 = 4. Next, a₃ = 4 + 2(3)/2 + 1 = 4 + 3 + 1 = 8. Then, a₄ = 8 + 3(4)/2 + 1 = 8 + 6 + 1 = 15. Finally, a₅ = 15 + 4(5)/2 + 1 = 15 + 10 + 1 = 26. Thus option B is correct. The successive increases are 2, 4, 7, and 11, so this is not a constant-difference progression. Omitting the final +1, treating the triangular expression as fixed, or using the wrong index can lead to the distractors 24, 28, or 30.

Correct answer: B. 15

Explanation: A recursive rule must be applied one step at a time. To obtain a₂, use n = 1: a₂ = 2 + 1(2)/2 + 1 = 2 + 1 + 1 = 4. To obtain a₃, use n = 2: a₃ = 4 + 2(3)/2 + 1 = 4 + 3 + 1 = 8. To obtain a₄, use n = 3: a₄ = 8 + 3(4)/2 + 1 = 8 + 6 + 1 = 15. Therefore option B is correct. The successive increments are 2, 4, and 7, so this is not an arithmetic progression with a fixed common difference. Option A could arise from omitting the final 1, while options C and D reflect incorrect substitution or arithmetic. The changing value of n is central to the calculation.

Correct answer: A. 67

Explanation: Substituting \(n=1,2,3\) in the recursive rule gives subtractions of \(2,4,7\), respectively. Thus, \(a_2=80-2=78\), \(a_3=78-4=74\), and \(a_4=74-7=67\). Therefore, the correct answer is 67. A value such as 69 can result from incorrectly subtracting 5 instead of 7 in the final step. Exam tip: to find \(a_4\), apply the recurrence only for \(n=1\) through \(n=3\).

Correct answer: B. 74

Explanation: 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.

Correct answer: C. \(\frac{17}{2}\)

Explanation: 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.

Correct answer: B. 10

Explanation: 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.

Correct answer: D. 31

Explanation: 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.

Correct answer: B. 13

Explanation: 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.

Correct answer: B. 25

Explanation: 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.

Correct answer: D. 48

Explanation: 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.

Correct answer: B. वह अनुक्रम जिसमें प्रत्येक नया पद पिछले एक या अधिक पदों के आधार पर ज्ञात हो

Explanation: 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.

Correct answer: B. 13

Explanation: 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.

Correct answer: C. 38

Explanation: 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.

Correct answer: B. 8th

Explanation: 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.

Correct answer: D. 48

Explanation: 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.

Correct answer: C. 16

Explanation: 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.

Correct answer: C. 41

Explanation: 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.

Correct answer: C. 24

Explanation: 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.

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

Explanation: 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.

Correct answer: C. 36

Explanation: 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 \(a_{n+1}\).

Correct answer: C. 19

Explanation: 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.

Correct answer: B. 117

Explanation: 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.

Correct answer: C. 132

Explanation: 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.

Correct answer: B. 31

Explanation: At each step, the recursive rule adds \(n^2+2n+1=(n+1)^2\). Thus, \(a_2=2+4=6\), \(a_3=6+9=15\), and \(a_4=15+16=31\). Therefore, 31 is correct. Option 30 may result from incorrectly using 15 instead of 16 for the final increment. Exam tip: quickly recognise \(n^2+2n+1\) as \((n+1)^2\).