Recursive rule

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.

(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.

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

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

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.

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.

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.

The governing concept is the recursive rule of a geometric progression: each new term equals the preceding term multiplied by the common ratio. Start with a₁ = 3. Then a₂ = 3 × (−2) = −6, a₃ = −6 × (−2) = 12, and a₄ = 12 × (−2) = −24. Thus the first four terms are (3, −6, 12, −24), making option B correct. The negative ratio explains the alternating signs, while the magnitude doubles at each step. Option A incorrectly uses a positive ratio of 2. Option C treats the ratio as though it were the second term, and option D changes the given first term from 3 to −3. The recursive calculation confirms every entry.