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SubjectsMathematics
पुनरावर्ती नियम
In Class 9 Mathematics, the topic Recursive Rule in Sequences and Progressions explains how a sequence can be defined by giving one or more starting terms and a rule that uses earlier terms to find the next one. Students learn to read and write such rules, generate sequence terms step by step, recognize patterns, and check whether a rule correctly describes a sequence. The topic also connects recursive descriptions with familiar arithmetic and geometric progressions, helping students understand how terms change and how sequence patterns can be represented mathematically.
Correct answer: C. \(22\)
Explanation: The recursive rule increases each term by 4. Thus, \(a_2=3+4=7\), \(a_3=11\), and \(a_4=15\). Therefore, \(a_2+a_4=7+15=22\), so option C is correct. \(20\) can result from an incorrect use of the common increase. Exam tip: write the required terms in order before finding their sum.
Correct answer: C. 16
Explanation: Given a_1=2, the recursive rule gives a_2=a_1^2=2^2=4. Then a_3=a_2^2=4^2=16. Hence, 16 is correct. The value 8 is the cube of 2, but this rule requires squaring each preceding term. Exam tip: in a recursive sequence, apply the rule step by step until the required term is reached.
Correct answer: C. 5
Explanation: Given \(a_1=1\), \(a_2=a_1^2+1=1^2+1=2\). Then \(a_3=a_2^2+1=2^2+1=5\). Therefore, the correct answer is 5. Option 4 is only \(2^2\); the recursive rule also requires adding 1. Exam tip: for every new term, square the preceding term and then add 1.
Correct answer: B. 70
Explanation: The recursive rule subtracts 15 from each preceding term. Thus, \(a_2=100-15=85\), and then \(a_3=85-15=70\). Option 85 is a close distractor because it is the second term, not the third term. Exam tip: write each step in a recursive sequence until you reach the required term.
Correct answer: C. 27
Explanation: The recursive rule says to add 6 to the previous term each time. Thus, a_2=15, a_3=21, and a_4=27. The value 21 is the third term, so it is a close but incorrect option. Exam tip: from a_1 to a_4, apply the rule three times.
Correct answer: C. 32
Explanation: The recursive rule makes each new term four times the preceding term. Thus, a_2=4×2=8 and a_3=4×8=32. Therefore, 32 is correct. The value 64 would be a_4, not a_3. Exam tip: to find a_3 from a_1, apply the rule twice.
Correct answer: A. \(a_1=4,\ a_{n+1}=a_n+6\)
Explanation: In option A, each new term is formed by adding 6 to the previous term, so \(a_{n+1}-a_n=6\) is constant. This is the defining property of an AP. In option D, the difference changes with \(n\). Exam tip: check for a constant difference.
Correct answer: C. 21
Explanation: Put \(n=1\) in the recursive rule: \(a_2=\frac{a_1}{2}+1\). Therefore, \(a_2=\frac{40}{2}+1=20+1=21\). Option 20 only halves 40; the rule also requires adding 1. Exam tip: to find \(a_{n+1}\), substitute the given term first and then follow the order of operations.
Correct answer: D. 19
Explanation: In a recursive rule, the value of n changes for each new term. Thus, a_2=1+3(1)=4, a_3=4+3(2)=10, and a_4=10+3(3)=19. Therefore, the correct answer is 19. The value 16 would result from incorrectly adding only 6 in the last step. Exam tip: To find a_4, apply the rule successively for n=1, 2, and 3.
Correct answer: B. 17
Explanation: For n=1, a_2=25-(2×1+1)=22. Then, for n=2, a_3=22-(2×2+1)=22-5=17. Hence, 17 is correct. A value such as 19 can result from incorrectly using a number other than 5 in the second step. Exam tip: increase the value of n by 1 for each successive term.
Correct answer: C. 9
Explanation: The second term is 5, and each following term is 2 more than the preceding term. Thus, \(a_3=5+2=7\) and \(a_4=7+2=9\). Therefore, the correct answer is 9. Note that 7 is \(a_3\), not \(a_4\). Exam tip: In a recursive rule, apply the stated operation step by step from the given term.
Correct answer: C. 16
Explanation: By the recursive rule, each new term is twice the preceding term. Thus the sequence is \(1, 2, 4, 8, 16\), so \(a_5=16\). The value \(8\) is the fourth term, \(a_4\), not the fifth term. Exam tip: write the term numbers while extending a recursively defined sequence.
Correct answer: D. (34)
Explanation: A recursive rule defines a term by using earlier terms. Here, every term after the first two is found by adding the two immediately preceding terms. The starting values are already given, so the sequence can be built one position at a time. Keeping the position labels visible helps prevent skipping a term or confusing the required term with the previous one.
Start with \(a_1=3\) and \(a_2=5\). Then \(a_3=3+5=8\), \(a_4=5+8=13\), \(a_5=8+13=21\), and \(a_6=13+21=34\). Therefore the correct choice is D, \(34\). The other numerical choices result from stopping too early or making an addition error.
Correct answer: C. 16
Explanation: The governing concept is a recursive rule, in which each term is obtained from the preceding term. Examine the consecutive changes in the sequence: 7 − 4 = 3, 10 − 7 = 3, and 13 − 10 = 3. Since the same difference occurs each time, the recursive rule is “add 3 to the previous term.” Applying this rule to the last known term gives 13 + 3 = 16. Therefore, option C is correct. This sequence is also an arithmetic progression with common difference 3, although the question specifically highlights the step-by-step recursive construction. Option A, 14, changes by only 1; option B, 15, changes by 2; and option D, 17, changes by 4, so none preserves the established rule.
Correct answer: B. 2
Explanation: The governing pattern is a recursive operation applied to the previous term. Each listed term is one-half of the preceding term: 16 = 32 ÷ 2, 8 = 16 ÷ 2, and 4 = 8 ÷ 2. Continuing exactly the same rule, the next term is 4 ÷ 2 = 2. Therefore option B is correct. Option D merely repeats the last known term, while option C does not maintain the constant ratio. Option A would be obtained only after applying the division rule twice more: 4 becomes 2 and then 1. The sequence is also geometric with common ratio 1/2, but the essential task is to continue its recursive rule.
Correct answer: C. 6
Explanation: Check the difference between consecutive terms: \(12-6=6\), \(18-12=6\), and \(24-18=6\). Thus, each new term is obtained by adding \(6\) to the previous term, so the recursive rule is \(a_{n+1}=a_n+6\). The number \(12\) is the second term, not the amount added. Exam tip: to find the number added in a recursive rule, subtract one consecutive term from the next.
Correct answer: B. 3
Explanation: Check the differences between consecutive terms: \(15-18=-3\), \(12-15=-3\), and \(9-12=-3\). Therefore, 3 is subtracted from each previous term, so the recursive rule is \(a_{n+1}=a_n-3\). If 2 were subtracted, the next term would be 16, which does not match the sequence. Exam tip: find the difference between two consecutive terms to identify a recursive rule.
Correct answer: B. 3
Explanation: Each term is obtained by multiplying the previous term by 3: \(3\times3=9\), \(9\times3=27\), and \(27\times3=81\). Thus, the recursive rule is \(a_n=3a_{n-1}\), so the multiplier is 3. Although 9 is a term of the sequence, it is not the multiplier. Exam tip: divide a term by the preceding term, for example \(27\div9=3\), to find the multiplier.
Correct answer: B. 25
Explanation: By the recursive rule, each new term is obtained by adding 7 to the preceding term. Thus, \(a_2=11+7=18\) and \(a_3=18+7=25\). Therefore, the correct answer is 25. Option 18 is the second term, so it is a close but incorrect distractor. Exam tip: to find \(a_3\) from \(a_1\), apply the rule twice.
Correct answer: C. 32
Explanation: The recursive rule adds 9 to each preceding term. Thus, a_2=5+9=14, a_3=14+9=23, and a_4=23+9=32. Therefore, 32 is correct. Note that 23 is the third term, not the fourth. Exam tip: from a_1 to a_4, apply the rule three times.
Correct answer: B. (10)
Explanation: The terms are (90,30,10,\frac{10}{3}), so (a_4=\frac{10}{3}). None of the listed options is correct.
Correct answer: A. 10
Explanation: To evaluate a recursively defined sequence, apply the stated operation once for each transition between the starting index and the requested index. The sequence begins with a₁ = 90. Applying the rule once gives a₂ = 90 ÷ 3 = 30. Applying it a second time gives a₃ = 30 ÷ 3 = 10. Therefore option A is correct. Option C is the value of a₂, so it results from stopping one step too early. Option B does not arise from division by 3. Option D would be obtained by dividing a₃ by 3 once more, so it corresponds to a₄ rather than a₃. Correct indexing is essential in recurrence questions.
Correct answer: C. 11
Explanation: Apply the recursive rule successively for n=1, 2, and 3. Thus, a_2=2+1+1=4, a_3=4+2+1=7, and a_4=7+3+1=11. Therefore, the correct answer is 11. The value 10 may result from incorrectly treating the added quantity as fixed at every step. Exam tip: To find a_4, start from a_1 and apply the rule three times.
Correct answer: C. 13
Explanation: In the recursive rule, first put \(n=1\): \(a_2=a_1+2(1)+2=3+4=7\). Then put \(n=2\): \(a_3=a_2+2(2)+2=7+6=13\). Therefore, the correct answer is 13. The value 11 can result from not updating the value of \(n\) correctly in the second step. Exam tip: use the next integer value of \(n\) when finding each new term.
Correct answer: C. 26
Explanation: The governing concept is a second-order recursive sequence. Every term from the third onward equals the sum of the two terms immediately before it. The initial values are a₁ = 4 and a₂ = 6. Calculate in order: a₃ = a₂ + a₁ = 6 + 4 = 10; a₄ = a₃ + a₂ = 10 + 6 = 16; and a₅ = a₄ + a₃ = 16 + 10 = 26. Hence option C is correct. Option A is the fourth term, so it results from stopping one step too early. Options B and D cannot be obtained by adding the correct pair of preceding terms. The order matters: for a₅, the required pair is a₄ and a₃, not just one earlier term or the original starting values.
