Sequences

Correct answer: B. 35

Explanation: The given nth-term rule is \(a_n=4n+3\). Substituting \(n=8\), we get \(a_8=4\times8+3=32+3=35\). Therefore, 35 is correct. The value 37 would result if the constant added were 5, which is not the case here. Exam tip: For an nth-term question, first substitute the given term number for \(n\), then simplify.

Correct answer: C. 50

Explanation: To find the fourth term, substitute \(n=4\) in the rule: \(a_4=3(4)^2+2=3\times16+2=50\). Hence, 50 is correct. The nearby distractor 48 results from stopping after \(3\times16\) and not adding 2. In exams, substitute the value first and then follow the order of operations: power, multiplication, and addition.

Correct answer: C. 27

Explanation: Each successive term is 7 greater than the preceding term: 6, 13, 20, 27, 34. Therefore, the term after 20 is 20 + 7 = 27, so 27 is correct. Choosing 28 would make the difference from 20 equal to 8, which breaks the common pattern. Exam tip: Find the difference between consecutive terms to identify the rule of a sequence.

Correct answer: B. 86

Explanation: Each term in the sequence is 9 less than the preceding term: \(95-9=86\), and \(86-9=77\). Therefore, the missing value is 86. If 88 were used, the difference from 95 would be 7, so the common pattern would not hold. Exam tip: In a missing-term sequence, compare consecutive known terms first to identify the rule.

Correct answer: B. 48

Explanation: The differences between consecutive terms are \(8-3=5\), \(15-8=7\), \(24-15=9\), and \(35-24=11\). These are consecutive odd numbers, so the next difference is \(13\). Therefore, the next term is \(35+13=48\). Choosing 46 would repeat the difference 11, which does not follow the increasing-difference pattern. Exam tip: For such sequences, first list the consecutive differences and look for their pattern.

Correct answer: B. 75

Explanation: The successive subtractions are 3, 6, 9, and 12. Since each subtraction increases by 3, the next subtraction is 15. Therefore, the next term is \(90-15=75\). The value 72 would result from subtracting 18 next, which does not follow the given pattern. Exam tip: For such sequences, first find the differences between consecutive terms and then identify the pattern in those differences.

Correct answer: C. 1215

Explanation: In this sequence, each term is 3 times the preceding term: 5, 15, 45, 135, 405, 1215. Therefore, the sixth term is 1215. The value 405 is the fifth term, so it is a close but incorrect option. Exam tip: In a geometric sequence, check the common ratio between consecutive terms to find the next term.

Correct answer: C. 32

Explanation: In this sequence, each term is obtained by dividing the preceding term by 2: \(512\to256\to128\to64\). Therefore, the next term is \(64\div2=32\). Although 16 is related to the pattern, it comes after 32 and is therefore incorrect here. Exam tip: Compare consecutive terms to identify the rule of a sequence.

Correct answer: B. 81

Explanation: The terms are \(2^2,3^2,4^2,5^2,6^2,\ldots\). Hence, the \(n\)th term is \((n+1)^2\). Therefore, the \(8\)th term is \((8+1)^2=9^2=81\). Choosing \(64\) is a common error caused by taking \(8^2\), since the first term is \(2^2\), not \(1^2\). Exam tip: always check the offset between the term number and the base being squared.

Correct answer: A. 216

Explanation: The given terms are consecutive cubes: \(8=2^3\), \(27=3^3\), \(64=4^3\), and \(125=5^3\). Therefore, the next, or fifth, term is \(6^3=216\). \(225\) is not a cube number, so it cannot be the next term of this sequence. Exam tip: Rewrite sequence terms as cubes to identify the pattern in their bases.

Correct answer: C. 35

Explanation: The differences between consecutive terms are \(3,4,5,6\). Therefore, after the fifth term \(20\), the sixth term is \(20+7=27\), and the seventh term is \(27+8=35\). Hence, the correct answer is \(35\). Note that \(27\) is only the sixth term. Exam tip: write the successive differences to identify the next increment in such sequences.

Correct answer: C. 29

Explanation: From the third term onward, each term is the sum of the two immediately preceding terms: \(3+4=7\), \(4+7=11\), and \(7+11=18\). Therefore, the next term is \(11+18=29\). Although 27 may result from guessing a pattern in the differences, it does not follow the given sum rule. Exam tip: first check the relation among consecutive terms, then apply the same rule to find the next term.

Correct answer: B. 81

Explanation: In this sequence, each term is the sum of the two preceding terms: \(19=7+12\), \(31=12+19\), and \(50=19+31\). Therefore, the next term is \(31+50=81\). Although 79 can be obtained by adding 29 to the previous term, that rule does not fit all the terms. Exam tip: Along with checking differences, test whether each term is formed by adding the previous two terms.

Correct answer: B. fifth term

Explanation: Each term is obtained by multiplying the preceding term by 3: 4, 12, 36, 108, 324. Therefore, 324 is the fifth term. The fourth term is 108, so the fourth-term option is incorrect. Exam tip: List and count the terms in order to identify a term's position.

Correct answer: B. 8th term

Explanation: This is an arithmetic sequence with first term 8 and common difference 7. Its nth term is 8+(n-1)×7. Putting it equal to 57 gives (n-1)×7=49, so n=8. Therefore, 57 is the 8th term. The 7th term is 50, so it is not correct. Exam tip: To find the position of a number in a sequence, equate it to the nth-term expression.

Correct answer: B. 7th

Explanation: This is a decreasing arithmetic sequence in which each successive term is 6 less. The terms are 88, 82, 76, 70, 64, 58, 52. Therefore, 52 is the 7th term. The 6th term is 58, so option A is not correct. Exam tip: You can list the terms in order or use \(a_n=a+(n-1)d\).

Correct answer: A. \(a_n=6n-3\)

Explanation: The difference between consecutive terms is 6, so this is an arithmetic progression. Its nth term is \(a_n=a_1+(n-1)d=3+(n-1)\times6=6n-3\). In option B, substituting \(n=1\) gives 9, not the first term 3. Exam tip: always check a proposed rule by substituting \(n=1\).

Correct answer: A. \(a_n=9n-3\)

Explanation: This is an arithmetic sequence because 9 is added to each term. Here, \(a_1=6\) and the common difference is \(d=9\). Thus, \(a_n=a_1+(n-1)d=6+9(n-1)=9n-3\), so option A is correct. In option B, putting \(n=1\) gives 15 as the first term, so it is incorrect. Exam tip: check an \(a_n\) rule by substituting \(n=1\) to verify the first term.

Correct answer: C. (7, 12, 17, 22)

Explanation: For the first four terms, substitute n=1, 2, 3, and 4. This gives a_1=5(1)+2=7, a_2=12, a_3=17, and a_4=22. Hence, the correct sequence is (7, 12, 17, 22). Option D incorrectly treats 2 as the first term; it is obtained when n=0, whereas a sequence normally starts with n=1. Exam tip: always check the starting value of n before listing terms.

Correct answer: B. 40

Explanation: To find the fifth term, substitute \(n=5\) in the rule: \(a_5=5^2+3\times5=25+15=40\). Therefore, 40 is correct. A value such as 35 can result from an error in calculating \(5^2\) or \(3\times5\). Exam tip: after substituting the term number, evaluate powers first, then multiplication, and finally addition.

Correct answer: B. 50

Explanation: For the fourth term, substitute \(n=4\) in the rule: \(a_4=70-5(4)=70-20=50\). Hence, the correct answer is 50. The value 55 is obtained when \(n=3\), so it is the third term. Exam tip: In an nth-term question, first use the correct term number and then simplify carefully.

Correct answer: B. (4, 16, 36, 64)

Explanation: For the first term, put n=1: a_1=4(1)^2=4. Similarly, a_2=4(2)^2=16, a_3=4(3)^2=36, and a_4=4(4)^2=64. Hence, the correct sequence is (4, 16, 36, 64). Option C is only the sequence of squares; it has not been multiplied by 4. Exam tip: Substitute consecutive values of n and check at least the first four terms.

Correct answer: A. (2, 8, 26, 80)

Explanation: The general term is \(a_n=3^n-1\). Substituting \(n=1,2,3,4\) gives \(3^1-1=2\), \(3^2-1=8\), \(3^3-1=26\), and \(3^4-1=80\). Hence, the correct sequence is \((2, 8, 26, 80)\). Option B lists only the powers of 3 and does not subtract 1 from each term. Exam tip: To find initial terms from a general term, substitute consecutive values of \(n\), beginning with 1.

Correct answer: B. (3)

Explanation: Both sequences have common difference (4), so the term difference always remains (3). The (7)th terms also differ by (3).

Correct answer: A. The first is greater by (7)

Explanation: The direct answer is option A: the first sixth term is greater by 7. Both are arithmetic sequences. For the first sequence, \(a_1=12\), \(d=6\), so \(a_6=12+5(6)=42\). For the second, \(b_1=5\), \(d=6\), so \(b_6=5+5(6)=35\). Therefore \(a_6-b_6=42-35=7\). Option A states exactly this. Option B reverses the comparison, so it is wrong. Option C is wrong because 42 and 35 are not equal. Option D is wrong because the difference is 7, not 12; 12 is only the first term of the first sequence. Another quick method is to notice that corresponding terms have the same common difference, while the first terms differ by \(12-5=7\), so every corresponding pair differs by 7.