Recursive rule

Option B gives an initial term, \(a_1=4\), and forms each new term from the preceding term \(a_{n-1}\), so it is recursive. In A, C, and D, terms are obtained directly from \(n\). Exam tip: look for an initial value and a previous-term relation.

In option B, finding \(a_n\) requires \(a_{n-1}\), so it is recursive. For example, \(a_2=3+4=7\). In exams, look for a rule involving an earlier term.

The expression \\((-1)^n\\) alternates between -1 for odd n and +1 for even n. Since the recurrence uses this expression when moving from a_n to a_{n+1}, begin with a1 = 5 and evaluate the exponent at n = 1, 2, 3, and 4. This gives changes of -1, +1, -1, and +1. The terms are therefore 5, 4, 5, 4, 5, so a5 = 5 and option C is correct.

Step by step, a2 = a1 + (-1)^1 = 5 - 1 = 4. Next, a3 = 4 + (-1)^2 = 4 + 1 = 5. Then a4 = 5 + (-1)^3 = 4, and a5 = 4 + (-1)^4 = 5. The exponent is the current index n, not the new index n+1; confusing these would shift the signs. The supplied answer C and explanation correctly handle the alternating signs.

In the recursive rule, the sign of \((-1)^n\) alternates. For \(n=1\), \(a_2=4+2(-1)=2\); for \(n=2\), \(a_3=2+2(1)=4\); and for \(n=3\), \(a_4=4+2(-1)=2\). Therefore, the correct answer is \(2\). The value \(4\) is the third term, not the fourth term. Exam tip: In recursive-sequence questions, calculate terms one step at a time up to the required term.

Putting n=1 gives a_2=4+2(-1)^1=4-2=2. Then, putting n=2 gives a_3=2+2(-1)^2=2+2=4. Therefore, the correct answer is 4. Option 2 is the value of a_2, not a_3. Exam tip: in a recursive rule, substitute the correct value of n at each step.

The recursive rule adds successive powers of 2 to the current term. Thus, a_2=1+2^1=3, a_3=3+2^2=7, a_4=7+2^3=15, and a_5=15+2^4=31. Therefore, 31 is correct. The value 35 would require adding 20 instead of 16 in the last step, which does not follow the rule. Exam tip: to find a_5, apply the recurrence only for n=1 through n=4.

Each new term is obtained from the preceding term using the recursive rule. Thus,

a_2=100-2^1=98,

a_3=98-2^2=94, and

a_4=94-2^3=86.

Therefore, the correct answer is 86. A common error is to get 84 by not subtracting the powers of 2 in the correct order. Exam tip: to find a_4, apply the rule for n=1, 2, and 3.

Apply the recursive rule using the current value of n at each step. a_2=3+1(2)=5, a_3=5+2(3)=11, and a_4=11+3(4)=23. Therefore, the correct answer is 23. The value 21 would result from incorrectly taking the final increment as 10; for n=3, the increment is 3×4=12. Exam tip: To find a_4, apply the rule successively for n=1, 2, and 3.

In the recursive rule, subtract \(n(n+1)\) from the current term at each step. \(a_2=60-1\times2=58\), \(a_3=58-2\times3=52\), and \(a_4=52-3\times4=40\). Therefore, the correct answer is 40. Option 42 may seem close, but the third step requires subtracting \(3\times4=12\), not 10. Exam tip: to find \(a_4\), apply the rule successively for \(n=1,2,3\).

Given \(a_1=2\), we have \(2a_1=4\). Thus, 4 is added to each successive term: \(a_2=2+4=6\), \(a_3=6+4=10\), and \(a_4=10+4=14\). Therefore, the correct answer is \(14\). Getting \(16\) would mean adding 4 one extra time. Exam tip: In a recursive rule, substitute the first-term value and calculate the terms in order.

In the recursive rule, first use \(n=1\) to find \(a_2\): \(a_2=a_1+a_1+1=6+6+1=13\). Then use \(n=2\): \(a_3=a_2+a_1+2=13+6+2=21\). Therefore, the correct answer is 21. Option 20 could result from incorrectly using \(n=1\) again in the second step. Exam tip: increase the value of \(n\) by 1 for each next term.

Using the recursive rule, add the term’s index to the preceding term. Thus, \(a_3=3+3=6\), \(a_4=6+4=10\), and \(a_5=10+5=15\). Therefore, the correct answer is 15. The value 14 can result from incorrectly adding 4 instead of 5 in the final step. Exam tip: In a recursive sequence, write the terms step by step and use the correct index at each step.

By the recursive rule, each new term is obtained by adding its current index to the previous term. Thus, \(a_3=a_2+3=3+3=6\) and \(a_4=a_3+4=6+4=10\). Therefore, the correct answer is 10. The value 9 would result from adding 3 again after \(a_3\), but the rule requires adding 4 to find \(a_4\). Exam tip: calculate recursive sequences one term at a time up to the required term.

The recursive rule adds 4 to each previous term. To reach the seventh term from the first term, add 4 six times: \(a_7=5+6\times4=29\). Therefore, 29 is correct. The value 25 results from adding 4 only five times, so it is the sixth term. Exam tip: from \(a_1\) to \(a_n\), add the common increase \(n-1\) times.

The rule says that 6 is added to obtain every next term. Starting with a₁ = 12 gives a₂ = 18, a₃ = 24, a₄ = 30, a₅ = 36, and a₆ = 42. Equivalently, solve 12 + 6(n−1) = 42: 6(n−1) = 30, so n−1 = 5 and n = 6. Thus option B is correct; the other choices miscount the repeated additions.

Each term is 3 more than the preceding term. With \(a_1=5\), \(a_2=a_1+3=8\) and then \(a_3=11\), so B is correct. In exams, check the difference between the first two terms.

Given \(a_1=3\), we get \(a_2=2\times3+2=8\) and \(a_3=2\times8+2=18\). Hence, \(a_3-a_2=18-8=10\). Option 12 can result from an error while finding a term or subtracting. Exam tip: in a recursive sequence, calculate the required terms in order before finding their difference.

The recursive rule requires adding the previous two terms and then adding 1. Thus, a₃ = 5 + 2 + 1 = 8, a₄ = 8 + 5 + 1 = 14, and a₅ = 14 + 8 + 1 = 23. Therefore, 23 is correct. The value 22 would result from omitting the extra 1 in the final step. Exam tip: write the +1 separately at every step when using the rule.

By the recursive rule, each new term is the sum of the previous two terms plus 1. Thus, \(a_3=5+2+1=8\), and \(a_4=8+5+1=14\). Therefore, the correct answer is 14. A common error is to omit the extra 1, which can lead to 13. Exam tip: find \(a_3\) before calculating \(a_4\).

Apply the recursive rule to find the second term first: \(a_2=3\times4-2=10\). Then \(a_3=3\times10-2=28\). Therefore, 28 is correct. A result such as 26 can arise from an error in multiplication or subtraction in the second step. Exam tip: to find \(a_3\), apply the rule twice, starting from \(a_1\).