Sequences

Correct answer: D. 43

Explanation: The given rule is \(a_n=5n-2\). To find the ninth term, substitute \(n=9\): \(a_9=5\times9-2=45-2=43\). Therefore, the correct answer is 43. The value 41 may result from an error in multiplication or subtraction. Exam tip: While finding an \(n\)th term, first substitute the given value of \(n\) carefully.

Correct answer: A. 75

Explanation: The given rule is \(a_n=2n^2+3\). Substituting \(n=6\), we get \(a_6=2\times6^2+3=2\times36+3=75\). Therefore, the correct answer is 75. The value 72 results if one calculates \(2\times6^2\) but forgets to add 3. Exam tip: In such questions, square \(n\) first, then multiply and add.

Correct answer: B. 31

Explanation: Check the differences between consecutive terms: 15 − 7 = 8 and 23 − 15 = 8. Thus, this is an arithmetic sequence in which 8 is added each time. Therefore, the term after 23 is 23 + 8 = 31, and 31 + 8 = 39 confirms the pattern. Choosing 29 would not maintain the common difference. Exam tip: For a missing term in a sequence, first check the differences between consecutive terms.

Correct answer: C. 102

Explanation: This is a decreasing sequence in which each term is 8 less than the previous term: \(110-8=102\) and \(102-8=94\). Therefore, the blank is 102. If 101 were used, the decrease from 110 would be 9, so the common difference would not remain the same. Exam tip: In a missing-term sequence, check the differences using terms on both sides of the blank.

Correct answer: C. 49

Explanation: The given terms are consecutive square numbers: \(4=2^2\), \(9=3^2\), \(16=4^2\), \(25=5^2\), and \(36=6^2\). Therefore, the next term is \(7^2=49\). Although 47 is close, it is not a perfect square. Exam tip: in a sequence of squares, the base numbers increase by 1 each time.

Correct answer: B. 90

Explanation: The successive subtractions are 4, 8, 12, and 16. Since each subtraction increases by 4, the next subtraction is 20. Therefore, the next term is \(110-20=90\). Choosing 88 would require subtracting 22, which does not follow the pattern. Exam tip: write the differences between consecutive terms to identify the pattern quickly.

Correct answer: D. 2048

Explanation: Each term is 4 times the preceding term, so this is a geometric sequence. The sixth term is \(2\times4^{5}=2\times1024=2048\). Note that 1024 is the fifth term, making it a close but incorrect option. Exam tip: When consecutive terms have a common multiplier, use \(a_n=a_1r^{n-1}\).

Correct answer: B. 1

Explanation: Each term is obtained by dividing the preceding term by 5: \(625\div5=125\), \(125\div5=25\), and \(25\div5=5\). Therefore, the next term is \(5\div5=1\). Zero would result from subtracting 5 from 5, but the pattern here is division. Exam tip: Compare consecutive terms using division to identify a multiplication or division pattern.

Correct answer: B. 343

Explanation: This is a sequence of cube numbers: \(1=1^3\), \(8=2^3\), \(27=3^3\), \(64=4^3\), and \(125=5^3\). Hence, the \(n\)th term is \(n^3\). Therefore, the 7th term is \(7^3=343\). Although 216 is also a cube, it is \(6^3\), so it is the 6th term. Exam tip: Express the given terms as powers of natural numbers to identify the pattern quickly.

Correct answer: D. 45

Explanation: The successive differences are 3, 4, 5, 6, and so on. Therefore, adding 7 to 21 gives the sixth term 28; adding 8 gives the seventh term 36; and adding 9 gives the eighth term 45. Hence, 45 is correct. Note that 36 is the seventh term, not the eighth. Exam tip: For such sequences, first identify the pattern in the consecutive differences.

Correct answer: C. 45

Explanation: From the third term onward, each term is the sum of the two immediately preceding terms: \(5+6=11\), \(6+11=17\), and \(11+17=28\). Therefore, the next term is \(17+28=45\). An option such as 43 does not follow the rule of adding the previous two terms. Exam tip: For such sequences, first check relations such as addition, subtraction, or multiplication between consecutive terms.

Correct answer: D. 89

Explanation: From the third term onward, each term is the sum of the two immediately preceding terms: 21 = 8 + 13, 34 = 13 + 21, and 55 = 21 + 34. Therefore, the next term is 34 + 55 = 89. A number such as 84 may result from extending the differences incorrectly. Exam tip: Verify the same rule using at least two consecutive terms before choosing the answer.

Correct answer: B. 5th

Explanation: Each term in the sequence is 3 times the preceding term: 7, 21, 63, 189, 567. Therefore, 567 is the 5th term. Since 189 is the 4th term, the 4th-term option is incorrect. Exam tip: List the terms in order and count their positions carefully.

Correct answer: C. 10th term

Explanation: This is an arithmetic sequence with first term 12 and common difference 7. Its nth term is \(a_n=12+(n-1)\times7\). Putting \(a_n=75\), we get \(12+(n-1)\times7=75\), so \(n-1=9\) and \(n=10\). Therefore, 75 is the 10th term. The 9th term is \(68\), so option B is not correct. Exam tip: To find a term's position, identify the first term and common difference, then use \(a_n=a+(n-1)d\).

Correct answer: A. The statement is correct; adding 3 after 20 gives 23.

Explanation: The statement is correct because consecutive differences are \(5-2=3\), \(8-5=3\), and \(11-8=3\). Hence, the term after 20 is \(20+3=23\). Exam tip: check the common difference first.

Correct answer: A. \(a_n=7n-2\)

Explanation: The difference between consecutive terms is 7, so this is an arithmetic sequence. Its nth term is \(a_n=a_1+(n-1)d=5+(n-1)\times7=7n-2\). For \(a_n=7n+5\), substituting \(n=1\) gives 12, not the first term 5. Exam tip: check a proposed rule by substituting \(n=1\) for the first term.

Correct answer: A. \(a_n=10n-2\)

Explanation: This is an arithmetic sequence with first term \(a_1=8\) and common difference \(d=18-8=10\). Hence, \(a_n=a_1+(n-1)d=8+(n-1)10=10n-2\). Therefore, option A is correct. In option B, substituting \(n=1\) gives the first term as \(18\), not \(8\). Exam tip: Test a proposed nth-term rule by substituting \(n=1\) and \(n=2\) to check the first two terms.

Correct answer: B. (7, 13, 19, 25)

Explanation: For the first four terms, substitute n=1, 2, 3, and 4. This gives a_1=6(1)+1=7, a_2=13, a_3=19, and a_4=25. Hence, the sequence is (7, 13, 19, 25). Option C incorrectly treats 1 as the first term; when n starts at 1, the first term is 7. Exam tip: List n=1, 2, 3, 4 and substitute each value in the nth-term rule.

Correct answer: C. 60

Explanation: To find the sixth term, substitute n=6 in the rule: a_6=6^2+4(6)=36+24=60. Therefore, 60 is correct. A value such as 54 usually results from evaluating 6^2 incorrectly or making an addition error. In exams, calculate powers first, then multiplication, and finally addition.

Correct answer: C. 55

Explanation: To find the fifth term, substitute \(n=5\) in the rule: \(a_5=90-7(5)=90-35=55\). Therefore, 55 is correct. Choosing 52 results from an error in multiplication or subtraction. Exam tip: In an \(n\)th-term rule, substitute the value of \(n\) first, using brackets, and then simplify.

Correct answer: B. (5, 20, 45, 80)

Explanation: Substituting n = 1, 2, 3, and 4 in the rule gives 5(1)^2 = 5, 5(2)^2 = 20, 5(3)^2 = 45, and 5(4)^2 = 80. Hence, the sequence is (5, 20, 45, 80). Option A consists of multiples of 5, but it follows the rule 5n, not 5n². Exam tip: Substitute the first few natural-number values of n to match a sequence with its rule.

Correct answer: A. (2,14,62)

Explanation: Substitute n=1,2,3: a_1=4^1-2=2, a_2=4^2-2=14, and a_3=4^3-2=62. Hence, the correct sequence is (2,14,62). Option B lists only powers of 4 and does not subtract 2 from each term. Exam tip: Unless stated otherwise, begin finding sequence terms with n=1.

Correct answer: B. (7)

Explanation: Both sequences have common difference (5), so the term difference always remains (7). The (8)th terms also differ by (7).

Correct answer: A. The first is greater by (9)

Explanation: Both sequences have difference (7), and their first terms differ by (9). So their (7)th terms also differ by (9).

Correct answer: B. 135

Explanation: This is an arithmetic progression with first term 5 and common difference 7. Its first 6 terms are 5, 12, 19, 26, 33, and 40, whose sum is 135. A result such as 138 can arise from an error while adding the terms. Exam tip: You can also use \(S_n=\frac{n}{2}[2a+(n-1)d]\) directly for an arithmetic progression.