Recursive rule

Each new term is the sum of the two immediately preceding terms. Thus, \(a_3=8+3=11\), and \(a_4=11+8=19\). Therefore, the correct answer is 19. The value 16 would result from adding \(8+8\), but the recursive rule requires adding \(a_3\) and \(a_2\). Exam tip: find \(a_3\) first, then use it to calculate \(a_4\).

The recursive rule makes each new term one-third of the preceding term. Thus, \(a_2=\frac{45}{3}=15\) and \(a_3=\frac{15}{3}=5\). Option 15 is \(a_2\), not \(a_3\). Exam tip: to find \(a_3\) from \(a_1\), apply the rule twice.

Using the recursive rule, add 4 to each term: 5, 9, 13, 17, 21. Therefore, 21 is the 5th term. The 4th term is 17, so the fourth-term option is incorrect. Exam tip: list the terms with their positions in recursive-sequence questions.

The governing concept is locating a value in a recursively defined arithmetic sequence. The initial term is a₁ = 10, and each next term is obtained by adding 5 to the preceding term. Successive calculation gives a₁ = 10, a₂ = 15, a₃ = 20, a₄ = 25, and a₅ = 30. Therefore 30 is the fifth term, so option B is correct. The same result follows from the arithmetic-sequence formula aₙ = a₁ + (n−1)d, where d = 5. Setting the value equal to 30 gives 30 = 10 + 5(n−1), so 20 = 5(n−1), n−1 = 4, and n = 5. Option A is 25, the fourth term. Options C and D correspond to 35 and 40, which occur after 30 and therefore cannot be the requested position.

The governing concept is a recursive rule: it must identify the starting term and describe how each term is obtained from the immediately preceding term. The first term of the sequence is 12, so the initial condition must be a₁ = 12. Next, compare consecutive terms: 15 − 12 = 3, 18 − 15 = 3, and 21 − 18 = 3. The same difference is added at every stage, so the recurrence is aₙ₊₁ = aₙ + 3. Checking it gives a₂ = 12 + 3 = 15, a₃ = 15 + 3 = 18, and a₄ = 18 + 3 = 21. Thus option A is correct. Option B adds 4, option C starts with the wrong first term, and option D multiplies by 3, producing 12, 36, … rather than the given sequence.

A recursive rule must give the starting value and describe how to obtain each term from the preceding term. Here the first term is 36, and the differences are 31−36 = −5, 26−31 = −5, and 21−26 = −5. Therefore, subtracting 5 each time gives aₙ₊₁ = aₙ − 5. Option A has the wrong first term, B increases, and D multiplies.

A recursive rule tells us how to obtain each term from the term immediately before it. The first term here is 3. Comparing consecutive terms shows that 15 is obtained from 3 by multiplying by 5, 75 is obtained from 15 by multiplying by 5, and 375 is obtained from 75 by multiplying by 5. Therefore the same multiplication is repeated at every step.

In symbols, the starting value is written as \(a_1=3\), and the repeated operation is \(a_{n+1}=5a_n\). Thus option C gives both the correct first term and the correct rule. Option A uses a constant addition of 12, which fails because 15 to 75 is not an increase of 12. The ratio between successive terms is the useful clue.

A recursive rule must include the correct initial term as well as the operation used to produce the next term. The sequence begins with 72. Dividing 72 by 3 gives 24, and dividing 24 by 3 gives 8. Therefore each new term is one-third of the preceding term. The rule is not based on subtracting a fixed number, because the differences are 48 and 16, which are not equal.

The correct symbolic description is \(a_1=72\) and \(a_{n+1}=\frac{a_n}{3}\). This is option C. Option B uses the correct division operation but starts with 24, so it would produce 24, 8, and a later term instead of the stated sequence beginning at 72. Both the starting value and the recursive operation must match.

The governing concept is repeated use of a recursive relation. The condition a₁ = 20 gives the first term, and aₙ₊₁ = aₙ + 1 means that exactly 1 must be added whenever we move from one term to the next. To reach a₄ from a₁, there are three transitions, not four: a₂ = a₁ + 1 = 20 + 1 = 21; a₃ = a₂ + 1 = 21 + 1 = 22; and a₄ = a₃ + 1 = 22 + 1 = 23. Therefore option C is correct. The arithmetic-progression check gives a₄ = a₁ + (4 − 1)d = 20 + 3(1) = 23. Option A is the second term, option B is the third term, and option D results from adding 1 four times instead of the required three transitions.

By the recursive rule, each new term is 25 less than the previous term. Thus the terms are 100, 75, 50, 25. Therefore, a₄ = 25. The value 50 is the third term, so it is a close but incorrect option. Exam tip: to find a₄ from a₁, subtract 25 three times.

The recursive rule adds 6 to each preceding term. Thus the sequence is 4, 10, 16, 22, 28. Hence, \(a_2=10\) and \(a_5=28\), so \(a_5-a_2=28-10=18\). The value 24 would result from counting four common differences, but there are only three differences between \(a_2\) and \(a_5\). Exam tip: check the difference between the term indices when subtracting terms of an arithmetic sequence.

The recursive rule adds 5 to each preceding term. Thus, \(a_2=8+5=13\), \(a_3=18\), and \(a_4=23\). Therefore, \(a_2+a_4=13+23=36\). Option 41 would result from an incorrect calculation of the terms or common difference. Exam tip: in a recursive sequence, list the required terms in order before adding them.

Put \(n=1\) in the recursive rule: \(a_2=a_1^2=4^2=16\). Therefore, 16 is correct. The value 8 would be obtained by doubling 4, but the rule requires squaring the previous term. Exam tip: to find \(a_{n+1}\), apply the given rule directly to \(a_n\).

A recursive rule uses the previous term to find the next term. Here, \(a_2=a_1^2+1=3^2+1=9+1=10\). Therefore, 10 is correct. Option 9 is only \(3^2\); the required addition of 1 has not been made. Exam tip: substitute the given term into the rule first, then follow the order of operations.

By the recursive rule, each new term is 30 less than the preceding term. Thus, \(a_2=90-30=60\) and \(a_3=60-30=30\). Therefore, the correct answer is 30. Note that 60 is \(a_2\), not \(a_3\). Exam tip: in a recursive sequence, apply the rule step by step until the required term is reached.

The recursive rule adds 9 to each preceding term. Thus, a_2=25, a_3=34, and a_4=43. The value 45 does not follow from adding 9 successively to 16. Exam tip: to go from a_1 to a_4, apply the rule 3 times.

The recursive rule says that each new term is 6 times the preceding term. Thus, \(a_2=6\times2=12\) and \(a_3=6\times12=72\). Therefore, 72 is correct. The value 48 would result from multiplying 12 by 4 in the second step, which does not follow the given rule. Exam tip: to find \(a_3\), apply the recursive rule twice starting from \(a_1\).

In an arithmetic progression, the difference between consecutive terms is constant. Here, \(a_{n+1}-a_n=4\), so 4 is added each time. \(4a_n\) multiplies terms instead. Exam tip: identify the constant difference.

To find \(a_2\), use the recursive rule with \(n=1\): \(a_2=\frac{a_1}{2}+3=\frac{50}{2}+3=25+3=28\). Therefore, 28 is correct. Option 25 is only half of 50; the required addition of 3 has not been made. Exam tip: substitute the previous term into the recursive rule to obtain the next term.

In the recursive rule, use the index of the current term at each step. Thus, \(a_2=1+5(1)=6\), \(a_3=6+5(2)=16\), and \(a_4=16+5(3)=31\). Therefore, 31 is correct. The answer 26 may result from incorrectly adding 5 at every step. Exam tip: to find \(a_4\), substitute \(n=1,2,3\) successively in the rule.