Recursive rule

Correct answer: B. 67

Explanation: Under the recursive rule, each new term is formed using the previous two terms and the square of its own index. Thus, (Q_3=1+2+3^2=12), (Q_4=12+2+4^2=30), and (Q_5=30+12+5^2=67). Therefore, 67 is correct. A value such as 61 can result from using one of the preceding terms incorrectly. Exam tip: add the square of the current index at every step.

Correct answer: D. 46

Explanation: In a recursive rule, each new term is calculated from the preceding term. First, \(U_2=U_1^2-2=3^2-2=7\). Then, \(U_3=U_2^2-3=7^2-3=49-3=46\). Therefore, the correct answer is 46. A value such as 44 could result from subtracting 5 instead of 3 in the final step. Exam tip: write the value of \(n\) separately at every step before subtracting it.

Correct answer: A. 47

Explanation: In the recursive rule, use the exponent corresponding to the current term. Thus, \(S_2=75-2^2=71\), \(S_3=71-2^3=63\), and \(S_4=63-2^4=47\). Therefore, 47 is correct. The value 49 results if 2 is incorrectly subtracted instead of \(2^2\) while finding \(S_2\). Exam tip: To reach \(S_4\), substitute \(n=2,3,4\) in order.

Correct answer: C. \(53\)

Explanation: Using the recursive rule, first put \(n=2\): \(T_2=(2+2)\times2+2=10\). Next, put \(n=3\): \(T_3=(3+2)\times10+3=53\). Therefore, the correct answer is \(53\). A value such as \(49\) can result from using an incorrect multiplier instead of \(n+2\). Exam tip: substitute the value of \(n\) into the rule first, then use the preceding term.

Correct answer: A. समांतर श्रेणी

Explanation: An arithmetic progression has a constant difference between consecutive terms. Its recursive form is \(a_n=a_{n-1}+d\), so the same fixed number \(d\) is added each time. A geometric progression uses a constant multiplier, not a fixed addition. Exam tip: “constant difference” identifies an AP.

Correct answer: D. 28

Explanation: The governing concept is a recursive sequence in which later terms are obtained by adding a quantity depending on n. To connect a₂ with a₅, use the recurrence for n = 3, 4 and 5. The increments are 4(3) − 2 = 10, 4(4) − 2 = 14 and 4(5) − 2 = 18. Therefore a₅ = a₂ + 10 + 14 + 18 = a₂ + 42. Given a₅ = 70, a₂ = 70 − 42 = 28. Thus option D is correct. A forward check gives 28 + 10 = 38, 38 + 14 = 52 and 52 + 18 = 70. The other options result from using the wrong range of n, omitting an increment or making an arithmetic subtraction error.

Correct answer: A. 4

Explanation: Using the recursive rule, (b_2=3b_1+2). Therefore, (b_3=3(3b_1+2)+2=9b_1+8). Since (b_3=44), we get (9b_1+8=44), so (9b_1=36) and hence (b_1=4). Substituting 5, 6, or 7 would not give (b_3=44). Exam tip: expand a recursive relation step by step until it connects to the required term.

Correct answer: C. (c_2=5) and (c_3=11)

Explanation: In a recursive rule, each new term is obtained from the preceding term. For n=2, c_2=c_1+2(2-1)=3+2=5. Then, for n=3, c_3=c_2+3(3-1)=5+6=11. Hence, option C is correct. In option B, c_2 is correct, but c_3 is calculated incorrectly because 6 must be added. Exam tip: substitute the value of n separately at every step and use the previous term carefully.

Correct answer: D. (6)

Explanation: (d_5=58) and (d_6=82), so the value first exceeds (60) at (n=6). In exams, choose the first term that becomes greater.

Correct answer: B. (14)

Explanation: (e_3=9) and (e_4=23), so the difference is (14). In exams, find both terms correctly before taking the difference.

Correct answer: A. (54)

Explanation: The first four terms are (3), (8), (16), and (27), and their sum is (54). In exams, add all required terms for a sum.

Correct answer: D. (47:23)

Explanation: Using the recursive rule successively, we get g_2=5, g_3=11, g_4=23, and g_5=47. Therefore, g_5:g_4=47:23, so (47:23) is correct. In (46:23), the second term is correct but the first term is off by 1. Exam tip: calculate the terms in order up to the required position and do not reverse the ratio.

Correct answer: C. 67

Explanation: Apply the recursive rule step by step: \(h_3=2(8)-3+2(3)=19\). Next, \(h_4=2(19)-8+2(4)=38\), and \(h_5=2(38)-19+2(5)=67\). Therefore, the correct answer is \(67\). A value such as \(63\) can result from using the wrong value of \(n\) in \(2n\) or mishandling the subtraction of \(h_{n-2}\). Exam tip: substitute the value of \(n\) separately at every step.

Correct answer: B. 74

Explanation: In the recursive rule, substitute the current value of n at each step. Thus, p_2=2+(2^2+2+1)=9, p_3=9+13=22, p_4=22+21=43, and p_5=43+31=74. Therefore, the correct answer is 74. A value such as 68 can result from an incorrect term calculation. Exam tip: to move from p_1 to p_5, add the terms corresponding only to n=2,3,4,5.