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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: B. 36
Explanation: The recursive rule adds 5 to each preceding term. Thus, \(a_2=8+5=13\), \(a_3=18\), and \(a_4=23\). Therefore, \(a_2+a_4=13+23=36\). Option 41 would result from an incorrect calculation of the terms or common difference. Exam tip: in a recursive sequence, list the required terms in order before adding them.
Correct answer: C. 16
Explanation: Put \(n=1\) in the recursive rule: \(a_2=a_1^2=4^2=16\). Therefore, 16 is correct. The value 8 would be obtained by doubling 4, but the rule requires squaring the previous term. Exam tip: to find \(a_{n+1}\), apply the given rule directly to \(a_n\).
Correct answer: C. 10
Explanation: A recursive rule uses the previous term to find the next term. Here, \(a_2=a_1^2+1=3^2+1=9+1=10\). Therefore, 10 is correct. Option 9 is only \(3^2\); the required addition of 1 has not been made. Exam tip: substitute the given term into the rule first, then follow the order of operations.
Correct answer: B. 30
Explanation: By the recursive rule, each new term is 30 less than the preceding term. Thus, \(a_2=90-30=60\) and \(a_3=60-30=30\). Therefore, the correct answer is 30. Note that 60 is \(a_2\), not \(a_3\). Exam tip: in a recursive sequence, apply the rule step by step until the required term is reached.
Correct answer: C. 43
Explanation: The recursive rule adds 9 to each preceding term. Thus, a_2=25, a_3=34, and a_4=43. The value 45 does not follow from adding 9 successively to 16. Exam tip: to go from a_1 to a_4, apply the rule 3 times.
Correct answer: D. 72
Explanation: The recursive rule says that each new term is 6 times the preceding term. Thus, \(a_2=6\times2=12\) and \(a_3=6\times12=72\). Therefore, 72 is correct. The value 48 would result from multiplying 12 by 4 in the second step, which does not follow the given rule. Exam tip: to find \(a_3\), apply the recursive rule twice starting from \(a_1\).
Correct answer: A. \(a_{n+1}=a_n+4\)
Explanation: In an arithmetic progression, the difference between consecutive terms is constant. Here, \(a_{n+1}-a_n=4\), so 4 is added each time. \(4a_n\) multiplies terms instead. Exam tip: identify the constant difference.
Correct answer: C. 28
Explanation: To find \(a_2\), use the recursive rule with \(n=1\): \(a_2=\frac{a_1}{2}+3=\frac{50}{2}+3=25+3=28\). Therefore, 28 is correct. Option 25 is only half of 50; the required addition of 3 has not been made. Exam tip: substitute the previous term into the recursive rule to obtain the next term.
Correct answer: C. 31
Explanation: In the recursive rule, use the index of the current term at each step. Thus, \(a_2=1+5(1)=6\), \(a_3=6+5(2)=16\), and \(a_4=16+5(3)=31\). Therefore, 31 is correct. The answer 26 may result from incorrectly adding 5 at every step. Exam tip: to find \(a_4\), substitute \(n=1,2,3\) successively in the rule.
Correct answer: B. \(31\)
Explanation: In the recursive rule, first put \(n=1\): \(a_2=a_1-(1+3)=40-4=36\). Then put \(n=2\): \(a_3=a_2-(2+3)=36-5=31\). Therefore, the correct answer is \(31\). \(36\) is only the second term, not the third term. Exam tip: increase the value of \(n\) by one for each next term.
Correct answer: C. 22
Explanation: The recursive rule adds 4 to the previous term each time. Thus the sequence is 6, 10, 14, 18, 22, so (a_5=22). The value 20 is not obtained as the fourth term is 18. Exam tip: For a recursive sequence, list the terms in order and count their indices carefully.
Correct answer: C. 32
Explanation: The governing concept is a recursive sequence in which each term from the stated recurrence onward is twice the preceding term. Although a₁ = 1 is supplied, a₂ = 4 is also explicitly given, so use a₂ as the starting point for the recurrence. Calculate a₃ = 2a₂ = 2 × 4 = 8, a₄ = 2a₃ = 16, and a₅ = 2a₄ = 32. Therefore option C is correct. Option A is only a₄, so it stops one step early. The values 24 and 40 do not arise from repeatedly doubling the preceding term. The explicitly stated value a₂ also means there is no need to derive it from a₁.
Correct answer: C. 51
Explanation: By the recursive rule, each new term is the sum of the previous two terms. Thus, \(a_3=2+9=11\), \(a_4=9+11=20\), \(a_5=11+20=31\), and \(a_6=20+31=51\). Therefore, the correct answer is 51. The close distractor 31 is \(a_5\), not \(a_6\). Exam tip: Write the terms in order and check the term number carefully.
Correct answer: C. (31)
Explanation: This recurrence rule defines each term as the sum of the two terms immediately before it. The initial values are a1 = 2 and a2 = 9. Therefore, calculate in order: a3 is 9 + 2 = 11, a4 is 11 + 9 = 20, and a5 is 20 + 11 = 31. Thus the correct choice is option C. The order of the two previous terms does not change the sum, but their positions must be tracked correctly.
The safest method is to write every term with its index before continuing. The list is 2, 9, 11, 20, 31. In the final step, the two terms immediately before a5 are a4 = 20 and a3 = 11, so a5 = a4 + a3 = 20 + 11 = 31. A value such as 29 could come from an incorrect addition or from using the wrong earlier terms. The supplied answer C and explanation are accurate.
Correct answer: C. 31
Explanation: The governing idea is to compare consecutive terms in a recursively generated sequence. Each term increases by the same amount: 16−11 = 5, 21−16 = 5, and 26−21 = 5. Hence the next term is obtained by adding 5 to 26: 26 + 5 = 31. Option A is one less than the correct result, B does not use the constant difference, and D adds too much.
Correct answer: B. 5
Explanation: In this sequence, each term is half of the previous term: 80 becomes 40, 40 becomes 20, and 20 becomes 10. Therefore, the next term is \(10 \div 2=5\). Option 10 is incorrect because it does not apply the rule to the previous term. Exam tip: identify the operation between consecutive terms and apply it to the latest term.
Correct answer: B. 9
Explanation: Check the difference between consecutive terms: \(18-9=9\), \(27-18=9\), and \(36-27=9\). Thus, 9 is added to the previous term each time, so the recursive rule is \(a_{n+1}=a_n+9\). Although 18 is a term of the sequence, it is not the constant amount added each time. Exam tip: To find the number added in a recursive rule, subtract two consecutive terms.
Correct answer: C. 11
Explanation: Check the differences between consecutive terms: \(44-55=-11\), \(33-44=-11\), and \(22-33=-11\). Thus, each new term is obtained by subtracting 11 from the previous term: \(a_{n+1}=a_n-11\). Subtracting 10 would give 45 as the next term, which does not match the sequence. Exam tip: find the difference between consecutive terms to identify a recursive rule.
Correct answer: C. 6
Explanation: Check the ratio of consecutive terms: \(24\div4=6\), \(144\div24=6\), and \(864\div144=6\). Thus, each term is obtained by multiplying the previous term by \(6\), so the recursive-rule multiplier is 6. Option 8 may seem tempting because it appears in 864, but it is not the multiplier between consecutive terms. Exam tip: To find a multiplier, divide any term by the term immediately before it.
Correct answer: B. 34
Explanation: The recursive rule adds 8 to each preceding term. Thus, a_2=18+8=26 and a_3=26+8=34. Therefore, the correct answer is 34. The value 42 would result from adding 8 three times. Exam tip: to find a_3 from a_1, apply the rule twice.
Correct answer: C. 43
Explanation: The recursive rule adds 12 to each preceding term. Thus, \(a_2=7+12=19\), \(a_3=19+12=31\), and \(a_4=31+12=43\). Therefore, 43 is correct. Note that 31 is the value of \(a_3\), not \(a_4\). Exam tip: from \(a_1\) to \(a_4\), apply the rule three times.
Correct answer: C. 8
Explanation: By the recursive rule, each new term is obtained by dividing the previous term by 4. Thus, \(a_2=\frac{128}{4}=32\) and \(a_3=\frac{32}{4}=8\). Therefore, the correct answer is 8. Exam tip: to find \(a_3\) from \(a_1\), apply the recursive rule twice.
Correct answer: C. 36
Explanation: The governing concept is a recursive sequence in which each term is the sum of the previous two terms. Begin with a_1 = 2 and a_2 = 6. Then a_3 = 6 + 2 = 8, a_4 = 8 + 6 = 14, a_5 = 14 + 8 = 22, and a_6 = 22 + 14 = 36. Therefore, option C is correct. Option A is the fifth term, and option B can result from adding the wrong pair or stopping with an incorrect intermediate value. Option D is also inconsistent with the recurrence. Writing every intermediate term prevents an indexing error and confirms that the two terms immediately before a_6 are 22 and 14.
Correct answer: C. 22
Explanation: In the recursive rule, add the current index n and 2 at each step. Thus, \(a_2=10+1+2=13\), \(a_3=13+2+2=17\), and \(a_4=17+3+2=22\). Therefore, 22 is correct. The value 21 would result from incorrectly using \(n=2\) in the final step; moving from \(a_3\) to \(a_4\) requires \(n=3\). Exam tip: write the value of n separately for every successive term.
Correct answer: C. 17
Explanation: Using the recursive rule with n=1, a_2=a_1+1+2=10+3=13. Then, with n=2, a_3=a_2+2+2=13+4=17. Therefore, the correct answer is 17. The value 15 may result from incorrectly counting the increase in the second step. Exam tip: substitute the appropriate new value of n at every step of a recursive rule.
