Sequences

Correct answer: B. 175

Explanation: To find the seventh term, substitute n=7 in the rule: (a_7=4(7)^2-3(7)=4×49-21=196-21=175). Hence, the correct answer is 175. The value 196 is only 4×49; subtracting 3×7 is also necessary. Exam tip: evaluate the power first, then multiply and subtract.

Correct answer: C. 115

Explanation: Given \(a_n=n^3-2n\), substitute \(n=5\): \(a_5=5^3-2(5)=125-10=115\). Therefore, 115 is correct. A value such as 120 can result from not subtracting \(2n\) correctly. Exam tip: evaluate powers first, then multiplication, and finally subtraction.

Correct answer: C. 36

Explanation: Each term in the sequence is 9 greater than the previous term: 9, 18, 27, 36, 45. Therefore, the term after 27 is \(27+9=36\). Choosing 34 or 38 would not maintain the common difference of 9. Exam tip: Find the difference between consecutive terms to identify the pattern.

Correct answer: C. 166

Explanation: Check the differences between the given terms: \(152-138=14\) and \(138-124=14\). Thus, each successive term is 14 less than the previous term. Therefore, \(180-14=166\), so the blank is 166. If 164 were chosen, the next difference would be 12, so it would not maintain the pattern. Exam tip: In a missing-term sequence, first check the differences between consecutive given terms.

Correct answer: B. (70)

Explanation: The terms do not increase by a fixed amount, so we examine their consecutive differences. They are \(12-5=7\), \(22-12=10\), \(35-22=13\), and \(51-35=16\). These differences are 7, 10, 13, 16, and each is 3 greater than the previous one. This gives a clear rule for the next step.

The next difference is \(16+3=19\). Adding this to the fifth term gives \(51+19=70\). Therefore option B is correct. The sequence can also be understood as having second differences equal to 3, but calculating the first differences is enough here. The answer is not obtained by adding a constant number directly to the original terms.

Correct answer: A. 175

Explanation: The amounts subtracted are \(5, 10, 15, 20\). Each subtraction increases by \(5\), so the next subtraction is \(25\). Therefore, the next term is \(200-25=175\). \(170\) would result from subtracting \(30\), which does not follow the pattern. Exam tip: For next-term questions, first check the differences between consecutive terms.

Correct answer: D. 6144

Explanation: This is a geometric sequence in which each term is 4 times the preceding term. After the given terms, the 5th term is 384 × 4 = 1536, and the 6th term is 1536 × 4 = 6144. Therefore, 6144 is correct. The closest distractor, 1536, is the 5th term, not the 6th. Exam tip: identify the common multiplier to find subsequent terms quickly.

Correct answer: C. 9

Explanation: This is a geometric sequence in which each term is one-third of the preceding term: \(729\div3=243\), \(243\div3=81\), and \(81\div3=27\). Therefore, the next term is \(27\div3=9\). Neither 6 nor 12 follows the same division rule. Exam tip: Compare consecutive terms to identify the pattern or common ratio.

Correct answer: B. 121

Explanation: The given terms are consecutive perfect squares: \(16=4^2, 25=5^2, 36=6^2, 49=7^2\), and \(64=8^2\). Hence, the \(n\)th term is \((n+3)^2\). Therefore, the \(8\)th term is \((8+3)^2=11^2=121\). Note that \(144=12^2\) is the next, or \(9\)th, term. Exam tip: For such sequences, take square roots of the terms to identify the underlying number pattern.

Correct answer: B. 729

Explanation: The terms are \(4^3, 5^3, 6^3, 7^3, \ldots\), respectively. Hence, the fifth term is \(8^3=512\), while the sixth term is \(9^3=729\). Therefore, 729 is correct. In exams, first identify the sequence of base numbers and then apply the exponent.

Correct answer: A. \(3,7,11,15,\ldots\)

Explanation: In option A, consecutive differences are \(7-3=11-7=15-11=4\), so it is an AP. Its ratios are not constant, hence it is not a GP. Exam tip: check differences before ratios.

Correct answer: C. 62

Explanation: By the given rule, each new term is the sum of the two terms immediately before it: 14 = 4 + 10, 24 = 10 + 14, and 38 = 14 + 24. Therefore, the next term is 24 + 38 = 62. An option such as 60 does not equal the sum of the previous two terms. Exam tip: for this type of sequence, add the last two given terms to find the next term.

Correct answer: B. 5th term

Explanation: This is a geometric sequence in which each term is obtained by multiplying the previous term by 3: 9, 27, 81, 243, 729. Therefore, 729 is the 5th term. The 4th term is 243, so it is not correct. Exam tip: List the terms in order and check for a common multiplier.

Correct answer: C. 11th term

Explanation: This is an arithmetic sequence with first term 15 and common difference 9. Therefore, the nth term is \(a_n=15+(n-1)\times 9\). Putting \(a_n=105\), \(15+(n-1)\times9=105\), so \(n-1=10\) and \(n=11\). Hence, 105 is the 11th term. The 10th term is 96, so it is not correct. Exam tip: use \(a_n=a+(n-1)d\) to find the position of a term in an arithmetic sequence.

Correct answer: C. 9th term

Explanation: This is an arithmetic progression with first term 170 and common difference \(-12\). Therefore, \(a_n=170+(n-1)(-12)\). Setting \(a_n=74\) gives \(170-12(n-1)=74\), so \(n-1=8\) and \(n=9\). Hence, 74 is the 9th term. The 8th term is 86, so it is not correct. Exam tip: To find the position of a term, substitute its value in \(a_n=a+(n-1)d\).

Correct answer: A. \(a_n=11n-2\)

Explanation: The differences between consecutive terms are \(20-9=31-20=42-31=11\), so this is an arithmetic progression. Its nth term is \(a_n=a_1+(n-1)d=9+(n-1)\times11=11n-2\). In option B, putting \(n=1\) gives 20 as the first term, so it is incorrect. Exam tip: To check a linear rule quickly, substitute \(n=1\) and then \(n=2\).

Correct answer: A. \(a_n=9n+7\)

Explanation: This is an arithmetic sequence because each successive term increases by 9. Here, \(a_1=16\) and \(d=9\). Therefore, \(a_n=a_1+(n-1)d=16+9(n-1)=9n+7\), so option A is correct. In option B, substituting \(n=1\) gives 25 as the first term, not 16. Exam tip: Check a proposed \(a_n\) rule by putting \(n=1\) to verify the first term.

Correct answer: A. (3,11,19,27)

Explanation: Substituting \(n=1,2,3,4\) in the rule gives \(a_1=8(1)-5=3\), \(a_2=11\), \(a_3=19\), and \(a_4=27\). Therefore, the first four terms are \((3,11,19,27)\). The option \((8,16,24,32)\) uses only \(8n\) and ignores the subtraction of 5. Exam tip: start with \(n=1\) to find the first term.

Correct answer: C. 132

Explanation: Substitute n=6: a_6=3(6^2)+4(6)=3(36)+24=108+24=132. Therefore, the correct answer is 132. An option such as 128 can result from an error in squaring or multiplication. Exam tip: evaluate powers first, then multiply, and finally add.

Correct answer: C. 68

Explanation: To find the eighth term, substitute n=8 in the rule: \(a_8=140-9(8)=140-72=68\). Hence, 68 is correct. 66 may result from an incorrect subtraction of \(140-72\), while 72 is only the value of \(9\times8\), not the term itself. Exam tip: for an nth-term rule, multiply first and then subtract.

Correct answer: B. \(7, 28, 63, 112\)

Explanation: Given \(a_n=7n^2\), substitute \(n=1,2,3,4\). This gives \(7\times1^2=7\), \(7\times2^2=28\), \(7\times3^2=63\), and \(7\times4^2=112\). Hence, the sequence is \(7, 28, 63, 112\). Option A follows \(7n\), not \(7n^2\). Exam tip: For a rule-based sequence, substitute the first few natural-number values of \(n\).

Correct answer: A. (4,11,30,85)

Explanation: For the first four terms, take n=1,2,3,4. Thus, a_1=3^1+1=4, a_2=3^2+2=11, a_3=3^3+3=30, and a_4=3^4+4=85. Therefore, the correct sequence is (4,11,30,85). Option B contains only powers of 3 and does not add n. Exam tip: evaluate 3^n first and then add n for each term.

Correct answer: B. (4)

Explanation: Both sequences have common difference (7), so the term difference always remains (4). The (12)th terms also differ by (4).

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

Explanation: Both sequences have difference (8), and their first terms differ by (9). So their (9)th terms also differ by (9).

Correct answer: C. 360

Explanation: This is an arithmetic progression with first term \(a=8\), common difference \(d=8\), and \(n=9\) terms. Using \(S_n=\frac{n}{2}[2a+(n-1)d]\), we get \(S_9=\frac{9}{2}[16+8\times8]=\frac{9}{2}\times80=360\). Hence, 360 is correct. The value 384 may result from using an incorrect number of terms or last term. Exam tip: identify \(a\), \(d\), and \(n\) before applying the sum formula.