Recursive rule

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

The rule \(a_{n+1}=a_n+2n+1\) must be applied with the current value of \(n\) at each step. Start with \(a_1=3\). For \(n=1\), \(a_2=3+2(1)+1=6\). For \(n=2\), \(a_3=6+2(2)+1=11\). For \(n=3\), \(a_4=11+2(3)+1=18\). Finally, for \(n=4\), \(a_5=18+2(4)+1=27\). Thus option C is correct.

A common mistake is to use the same value of \(n\) repeatedly or to stop after calculating \(a_4\). The index in the recurrence identifies the step being performed, so it changes from 1 to 4. The resulting terms are \(3,6,11,18,27\), and the fifth term is therefore 27, not any of the other listed values.

In the recursive rule, the value of n changes at each step. Thus, a_2=40-3(1)=37, a_3=37-3(2)=31, and a_4=31-3(3)=22. Therefore, the correct answer is 22. Choosing 16 would result from incorrectly using a fixed subtraction instead of the changing value of 3n. Exam tip: to find a_4, apply the rule successively for n=1, 2, and 3.

This is a recursively defined sequence because each term is obtained from the preceding term rather than directly from its position. Begin with a₁ = 2 and apply the rule using the current value of n. For n = 1, a₂ = 2a₁ + 1 = 2(2) + 1 = 5. For n = 2, a₃ = 2a₂ + 2 = 2(5) + 2 = 12. For n = 3, a₄ = 2a₃ + 3 = 2(12) + 3 = 27. Thus the sequence starts 2, 5, 12, 27, and option C is correct. The other choices can result from omitting an index, using the wrong preceding term, or stopping the recurrence too early. The changing added value 1, 2, 3 is essential.

In a recursive rule, each new term is calculated from the preceding term. Thus, a_2=2(5)-2(1)=8, a_3=2(8)-2(2)=12, and a_4=2(12)-2(3)=18. Therefore, 18 is correct. A value such as 20 may result from forgetting to subtract 2n in the final step. Exam tip: to find a_4, substitute n=1, then 2, then 3 in order.