Recursive rule

Apply the recursive rule first with n=1: a_2=2+1(1+2)=5. Then use n=2: a_3=5+2(2+2)=5+8=13. Therefore, 13 is correct. A value such as 11 can result from calculating the second increment incorrectly. Exam tip: to find a_3, apply the rule successively for n=1 and n=2.

Using n=1, 2, and 3 in the recursive rule, the values subtracted are 1(3)=3, 2(4)=8, and 3(5)=15. Thus, a_2=100-3=97, a_3=97-8=89, and a_4=89-15=74. Therefore, 74 is correct. A value such as 76 would result from an incorrect final subtraction. Exam tip: to find a_4 from a_1, apply the rule three times.

Apply the recursive rule step by step: \(a_2=2(7)+3=17\), \(a_3=2(17)+3=37\), and \(a_4=2(37)+3=77\). Therefore, the correct answer is 77. An option such as 75 can result from an arithmetic error in the final step. In exams, write each previous term before calculating the next one.

Each new term is obtained by doubling the previous term and adding 3. Thus, \(a_2=2\times7+3=17\), and \(a_3=2\times17+3=37\). Therefore, 37 is correct. The value 35 would result from an incorrect calculation in the second step. Exam tip: in a recursive sequence, find the terms one step at a time rather than jumping directly to a later term.

In A, \(a_{n+1}-a_n=5\) for every \(n\), so the sequence is an AP. In B, the difference is \(a_n+5\), which is not fixed. Exam tip: check the difference of consecutive terms.

A recursive rule generates each term from the preceding one, so it needs \(a_1\) to begin. Then \(a_2=2a_1+5\). A common difference is relevant only for an arithmetic progression. Exam tip: always identify the starting value.

Each new term is obtained by adding the triangular quantity n(n+1)/2, where n is the index of the current term. To find a5, begin with a1 and perform the recurrence for n = 1, 2, 3, and 4. The added quantities are 1, 3, 6, and 10. Starting from 3, the terms become 4, 7, 13, and 23. Therefore a5 = 23, which is option B.

The calculation is a careful index exercise: a2 = 3 + 1(2)/2 = 4; a3 = 4 + 2(3)/2 = 7; a4 = 7 + 3(4)/2 = 13; and a5 = 13 + 4(5)/2 = 23. The increments are the triangular numbers 1, 3, 6, and 10, but they must be attached to the correct transition. The supplied answer B and explanation are consistent with the recurrence and contain no mathematical error.

Using \(n=1,2,3\), the successive amounts subtracted are \(1,3,6\). Thus, \(a_2=70-1=69\), \(a_3=69-3=66\), and \(a_4=66-6=60\). Therefore, the correct answer is 60. The value 58 would result from incorrectly taking the final decrement as 8 instead of 6. Exam tip: to find \(a_4\) from \(a_1\), apply the recursive rule three times.

Use the recursive rule first with \(n=1\): \(a_2=\frac{10}{2}+2(1)=7\). Then use \(n=2\): \(a_3=\frac{7}{2}+2(2)=\frac{15}{2}\). The value \(7\) is \(a_2\), not \(a_3\). Exam tip: while finding successive terms, substitute the correct index value of \(n\) each time.

To find \(a_2\), substitute \(n=1\) and \(a_1=12\) in the recursive rule: \(a_2=\frac{a_1}{2}+2(1)=\frac{12}{2}+2=8\). Hence, 8 is correct. The value 6 is only half of 12 and misses the \(2n\) term. Exam tip: when finding \(a_{n+1}\), first use the correct value of \(n\), then substitute the corresponding term \(a_n\).

Given \(a_1=6\), we have \(2a_1=12\). First, put \(n=1\): \(a_2=a_1+2a_1+1=6+12+1=19\). Next, put \(n=2\): \(a_3=a_2+2a_1+2=19+12+2=33\). Therefore, the correct answer is 33. The value 31 can result from forgetting to add \(n=2\) in the second step. Exam tip: in a recursive rule, substitute the correct value of \(n\) for each successive term.

Each new term is obtained by adding 5, so \(a_{n+1}-a_n=5\) is constant. Hence it is an arithmetic progression. A geometric progression requires a constant ratio. Exam tip: check consecutive differences first.

Using the recursive rule, first find the third term and then the fourth term. \(a_3=a_2+a_1+3=4+2+3=9\). Therefore, \(a_4=a_3+a_2+4=9+4+4=17\). Hence, the correct answer is 17. A value such as 16 can result from adding the \(+n\) term incorrectly. Exam tip: always substitute the current value of \(n\) separately for each new term.

Substitute \(n=3\) into the recursive rule: \(a_3=a_2+a_1+3=4+2+3=9\). Therefore, the correct answer is \(9\). Adding only \(a_2\) and \(a_1\) gives \(6\), but the extra index term \(3\) must also be included. Exam tip: always substitute the correct value of \(n\) when using a recursive rule.

By the recursive rule, each new term is obtained by squaring the previous term and subtracting that same previous term. Thus, \(a_2=3^2-3=6\) and \(a_3=6^2-6=30\). Therefore, the correct answer is 30. The value 36 is only \(6^2\); the rule also requires subtracting 6. Exam tip: always find \(a_2\) before calculating \(a_3\).

In a recursive rule, each new term is calculated from the preceding term. For


a_1=2,


a_2=a_1^2+a_1=2^2+2=6.

Then,


a_3=a_2^2+a_2=6^2+6=36+6=42.

Therefore, the correct answer is 42. The close distractor 36 is only


a_2^2=6^2

and misses the required addition of


a_2=6

in the rule. Exam tip: at every step, square the previous term and then add that same term.

The recursive rule increases each term by 3. To reach the eighth term, add 3 seven times to the first term: \(a_8=4+7\times3=25\). Therefore, 25 is correct. The value 22 is the seventh term, so it is a close but incorrect distractor. Exam tip: from \(a_1\) to \(a_n\), the number of increases is \(n-1\).

This is an arithmetic sequence because every term is obtained by adding the same number, 7, to the previous term. Starting at 9, the terms are 9, 16, 23, 30, 37, 44, and 51. The value 51 therefore appears as the seventh term, so option B is correct. The important point is that the first term already has index 1; it is not counted as a zero-th term.

Using the arithmetic-sequence formula gives the same result: a_n = a_1 + (n-1)d. Here, 51 = 9 + (n-1)7, so 42 = 7(n-1), which gives n-1 = 6 and n = 7. Listing the terms is often safer for a short question and confirms the result directly. The supplied answer B and its explanation correctly identify the seventh position. Option A would be one step too early and gives 44, not 51.

Using the recursive rule, \(a_2=2(4)+1=9\), \(a_3=2(9)+1=19\), and \(a_4=2(19)+1=39\). Therefore, \(a_4-a_2=39-9=30\). Option 28 may result from an error while generating the terms or subtracting them. Exam tip: apply a recursive rule step by step to each preceding term until the required term is reached.

Using the recursive rule, \(a_3=6+3+2=11\), \(a_4=11+6+2=19\), and \(a_5=19+11+2=32\). Therefore, the correct answer is 32. An answer such as 31 may result from adding the constant term 2 incorrectly. Exam tip: while finding each new term, include both previous terms and the given constant term.