Recursive rule

From the rule, \(a_{n+1}-a_n=2n\). The differences are 2, 4, 6, ... and are not constant, so the sequence is not an AP. Exam tip: write consecutive differences first to test for an AP.

Apply the recurrence successively, using n = 1 to calculate a₂ and n = 2 to calculate a₃. First, a₂ = a₁/2 + 3(1) = 100/2 + 3 = 53. Next, a₃ = a₂/2 + 3(2) = 53/2 + 6 = 26.5 + 6 = 32.5. Therefore option C is correct. The nearby integer choices result from mishandling the fraction or the index.

To find \(a_2\), substitute \(n=1\) and \(a_1=20\) into the recursive rule: \(a_2=\frac{20}{2}+3(1)=10+3=13\). Hence, 13 is correct. The value 11 would result from incorrectly adding 1 instead of \(3n\). Exam tip: when finding the next term from \(a_{n+1}\), substitute the correct value of \(n\) first.

Using the recursive rule with n=1, a_2=5+2(5)+1^2=16. Then, with n=2, a_3=16+2(5)+2^2=16+10+4=30. Therefore, the correct answer is 30. Option 29 is a close distractor, but it is 1 less than the value obtained when 2^2=4 is added correctly. Exam tip: For each next term, substitute the current value of n and keep a_1 fixed as 5.

The added term \(2n+1\) is the difference between consecutive squares: \((n+1)^2-n^2=2n+1\). Hence the terms are \(1,4,9,16,\dots\), or perfect squares. The difference is not constant, so it is not an AP. Exam tip: remember that square differences are consecutive odd numbers.

Using the recursive rule, first find the third term: \(a_3=a_2+a_1+2(3)=3+2+6=11\). Then \(a_4=a_3+a_2+2(4)=11+3+8=22\). Therefore, the correct answer is 22. A value such as 21 can result from an error while evaluating the \(2n\) term. Exam tip: write the current value of \(n\) explicitly before substituting it into a recursive rule.

In \(a_n=a_{n-1}+a_{n-2}\), each new term depends on two preceding terms, so both \(a_1\) and \(a_2\) are needed. The other rules use only \(a_{n-1}\), so one initial term is enough. Exam tip: count the earlier terms used in the rule.

Apply the recursive rule first to \(a_1=4\): \(a_2=4^2-3(4)=16-12=4\). Using \(a_2=4\) again, \(a_3=4^2-3(4)=4\). Hence, the correct answer is 4. The option 8 is incorrect because each term is obtained by subtracting three times the previous term from its square. Exam tip: To find \(a_3\), calculate \(a_2\) first.

Use the recursive rule step by step, finding each term from the preceding term. First, \(a_2=3^2-3+2=8\). Then \(a_3=8^2-8+2=64-8+2=58\). Therefore, the correct answer is 58. The value 64 is only \(8^2\); it ignores the \(-a_n+2\) part of the rule. Exam tip: To find \(a_3\), calculate \(a_2\) first rather than substituting \(a_1\) directly.

Put \(n=1\) in the recursive rule: \(a_2=a_1^2-a_1+2\). Hence, \(a_2=3^2-3+2=9-3+2=8\). Therefore, 8 is correct. Option 6 can result from an error in calculating \(3^2\) or in the final addition. Exam tip: to find \(a_2\), always substitute \(n=1\) in the recurrence relation.

For (a_{10}), (5) is added (9) times, so (7+45=52). Exam tip: the rule is applied one less time than the term number.

In a geometric sequence, each term is obtained by multiplying the preceding term by a fixed ratio. Here the ratio is 2, so \(a_{n+1}=2a_n\). Check: the second term is \(2\times5=10\). In exams, “twice” means multiply by 2, not add 2.

Using the recursive rule, \(a_2=2\times6+4=16\), \(a_3=2\times16+4=36\), and \(a_4=2\times36+4=76\). Hence, \(a_4-a_2=76-16=60\), so option D is correct. \(56\) results from an incorrect calculation of the terms. Exam tip: List the required terms in order before finding their difference.

Given \(a_1=6\), the recursive rule gives \(a_2=2\times6+4=16\) and \(a_3=2\times16+4=36\). Therefore, \(a_3-a_2=36-16=20\). Note that 16 is only the second term, not the required difference. Exam tip: in recursive-sequence questions, calculate the required terms in order before finding their difference.

By the recursive rule, each new term is obtained by adding the previous two terms and 4. Thus, \(a_3=5+9+4=18\), \(a_4=9+18+4=31\), and \(a_5=18+31+4=53\). Therefore, the correct answer is 53. A value such as 49 can result from missing the constant 4 in a step. Exam tip: write each term in order and include the stated constant at every step.

The recursive rule requires adding the previous two terms and then adding 4 each time. First, \(a_3=9+5+4=18\). Then, \(a_4=18+9+4=31\). Therefore, the correct answer is \(31\). The value \(29\) would result from adding only 2 instead of 4 in the final step, so it does not follow the given rule. Exam tip: write each intermediate term in order until you reach the required term.

The rule adds 4 to every preceding term, so the difference between consecutive terms is constantly 4. Hence it is an arithmetic progression. A difference of \(-4\) would make terms decrease. Exam tip: identify an AP by checking for a constant difference.

In option A, the difference between consecutive terms is \(a_{n+1}-a_n=d\), which is constant; hence it is an arithmetic progression. In option C, the difference \(n\) changes. Exam tip: check for a constant difference in an AP.

The governing concept is a recursive rule, so the sequence must be generated one transition at a time. The first term is a₁ = 4. To obtain a₂, use n = 1 in the increment: 7(1) - 2 = 5, giving a₂ = 4 + 5 = 9. Next, with n = 2, the increment is 7(2) - 2 = 12, so a₃ = 9 + 12 = 21. Finally, with n = 3, the increment is 7(3) - 2 = 19, giving a₄ = 21 + 19 = 40. Thus option D is correct. There are exactly three updates from a₁ to a₄. Using n = 4 before reaching a₄ or omitting an update would produce an incorrect distractor.

Apply the recursive rule using the correct value of \(n\) at each step. For \(n=1\), \(a_2=4+7(1)-2=9\). Then, with \(n=2\), \(a_3=9+7(2)-2=21\). Hence, 21 is correct. A value such as 20 can result from using an incorrect value for \(7n-2\) in the second step. Exam tip: use \(n=1\) to find \(a_2\), and \(n=2\) to find \(a_3\).