Sequences

Correct answer: C. (90)

Explanation: The first (6) terms are (4,9,14,19,24,29). Their sum is (99), so no listed option is correct.

Correct answer: B. 240

Explanation: Each successive term in the sequence decreases by 6. Therefore, the first 5 terms are 60, 54, 48, 42, and 36. Their sum is 60+54+48+42+36=240, so 240 is correct. Note that 230 is not the sum of the first four terms either; those terms add up to 204. Exam tip: Write out the required terms before adding them in a decreasing sequence.

Correct answer: C. 10th term

Explanation: This is an arithmetic progression with first term \(a=2\) and common difference \(d=5\). Its \(n\)th term is \(a_n=a+(n-1)d\). Thus, \(2+(n-1)\times5=47\), which gives \(n=10\). Therefore, 47 is the 10th term. The 9th term is \(42\), so it is not correct. Exam tip: To find the position of a given term, substitute it for \(a_n\) in \(a_n=a+(n-1)d\).

Correct answer: D. 48

Explanation: This is an arithmetic sequence in which each successive term increases by 4. Thus, the terms are 6, 10, 14, 18, 22, 26, \ldots Hence, \(a_5=22\) and \(a_6=26\). Therefore, \(a_5+a_6=22+26=48\), so option D is correct. The value 46 may result from calculating the sixth term incorrectly. Exam tip: while counting terms, treat the first term as \(a_1\).

Correct answer: C. 18

Explanation: This is an arithmetic progression with common difference \(d=21-15=6\). The third term is \(a_3=27\), and the sixth term is \(a_6=45\). Hence, \(a_6-a_3=45-27=18\). Option 12 represents the difference across only two terms, not three. Exam tip: use \(a_n-a_m=(n-m)d\) to find differences between terms quickly.

Correct answer: B. \(12\)

Explanation: This is an arithmetic progression with common difference \(4\). Here, \(a_2=14\) and \(a_5=10+4\times4=26\). Therefore, \(a_5-a_2=26-14=12\). The value \(14\) is \(a_2\), not the difference between the two terms. Exam tip: for the difference of terms, subtract the lower-indexed term from the higher-indexed term.

Correct answer: C. 39

Explanation: This is an arithmetic progression with first term 9 and common difference 5. To reach the seventh term, 5 is added six times: \(a_7=9+(7-1)\times5=39\). Therefore, 39 is correct. Getting 44 would mean adding 5 seven times, which gives the eighth term. Exam tip: use \(a_n=a_1+(n-1)d\) for an arithmetic progression.

Correct answer: B. 32

Explanation: This is an arithmetic sequence with first term \(a_1=72\) and common difference \(d=-8\). To reach the sixth term, 8 is subtracted 5 times: \(a_6=72+5(-8)=72-40=32\). Therefore, 32 is correct. The nearby distractor 40 results from subtracting 8 only 4 times. Exam tip: before the \(n\)th term, there are always \(n-1\) common differences.

Correct answer: C. 283

Explanation: The recursive rule is to multiply the previous term by 3 and then add 1. Therefore, the term after 94 is \(94\times 3+1=282+1=283\). Although 282 is three times 94, it misses the required addition of 1. Exam tip: for a recursive sequence, apply the stated rule directly to the last given term.

Correct answer: A. 2.125

Explanation: The rule is to divide the previous term by 2 and then subtract 1. Therefore, the term after 6.25 is \(6.25\div 2-1=3.125-1=2.125\). Option 3.125 results from dividing by 2 only; the subtraction of 1 is still required. Exam tip: In recursive sequences, apply the operations in the stated order—divide first, then subtract.

Correct answer: D. 50

Explanation: The differences between consecutive terms are 3, 6, 9, and 12. Since each difference increases by 3, the next difference is 15. Therefore, the next term is \(35+15=50\). Choosing 49 would give a difference of 14, which does not follow the pattern. Exam tip: For such sequences, first list consecutive differences and check their pattern.

Correct answer: A. 0

Explanation: The consecutive terms decrease by 10, 20, 30, and 40 respectively. Therefore, the next decrease is 50: \(50-50=0\). Hence, the next term is 0. Option 10 would result from subtracting 40 from 50, but 40 has already been used as the previous decrease. Exam tip: For such sequences, write the differences between consecutive terms and look for their pattern.

Correct answer: C. (240)

Explanation: This sequence also uses successive multiplication. The first term, 2, is multiplied by 2 to produce 4. Next, 4 is multiplied by 3 to produce 12, and 12 is multiplied by 4 to produce 48. The multiplying factors therefore follow the increasing pattern 2, 3, 4. The next factor should be 5.

Multiplying the last term by 5 gives \\(48\times5=240\\). Thus the next term is 240, which is option C. The sequence can also be described by \\(2\times1!, 2\times2!, 2\times3!, 2\times4!\\), producing 2, 4, 12, and 48; the next expression is \\(2\times5!=240\\). A value such as 288 would require a different rule not supported by the displayed pattern.

Correct answer: A. 3:5

Explanation: The fourth term of the first sequence is 192, and that of the second sequence is 320. Therefore, the ratio is 192:320. Dividing both terms by 64 gives 192:320 = 3:5, so option A is correct. The ratio 5:3 is the reverse ratio. Exam tip: While writing a ratio, keep the order of the first and second quantities unchanged.

Correct answer: B. 5

Explanation: This is an arithmetic progression with first term 8 and common difference 5. Its first 5 terms are 8, 13, 18, 23, and 28, and their sum is 8+13+18+23+28=90. The sum of the first 4 terms is only 62, so option 4 is not correct. Exam tip: For small numbers of terms, write the terms and add them to check quickly.

Correct answer: C. 34

Explanation: Each successive term in the sequence is obtained by adding 7: 6, 13, 20, 27, 34, 41. Therefore, the term immediately before 41 is 34. Option 36 is incorrect because adding 7 to 34 gives 41. Exam tip: Find the common difference between consecutive terms to identify a missing previous or next term quickly.

Correct answer: B. \(10\)

Explanation: Each term in this sequence is half of the preceding term: \(160, 80, 40, 20, 10, 5\). Therefore, \(10\) comes immediately before \(5\). Values such as \(8\) and \(12\) do not follow the halving rule. Exam tip: Check the ratio of consecutive terms to identify a sequence rule quickly.

Correct answer: A. \(4n+7\)

Explanation: This is an arithmetic sequence with first term \(a=11\) and common difference \(d=15-11=4\). Therefore, \(a_n=a+(n-1)d=11+(n-1)\times4=4n+7\). For \(4n+11\), substituting \(n=1\) gives 15, so it does not produce the first term 11. Exam tip: use \(a_n=a+(n-1)d\) after identifying the first term and common difference.

Correct answer: A. \(a_n=68-8n\)

Explanation: This is an arithmetic sequence with first term \(a_1=60\) and common difference \(d=52-60=-8\). Therefore, \(a_n=a_1+(n-1)d=60+(n-1)(-8)=68-8n\). In option A, substituting \(n=1\) gives 60 and \(n=2\) gives 52. Option B gives 52 when \(n=1\), so it is incorrect. Exam tip: verify a proposed nth-term rule by checking \(n=1\) and \(n=2\).

Correct answer: C. 35

Explanation: Each new term equals the sum of the two preceding terms. Thus, \(a_3=4+9=13\), \(a_4=9+13=22\), and \(a_5=13+22=35\). Therefore, 35 is correct. The value 31 can result from adding terms in the wrong order. Exam tip: Write every term sequentially in recursive-sequence questions.

Correct answer: B. 5

Explanation: This is a geometric sequence in which each term is 4 times the preceding term. Hence, \(a_n=3\times4^{n-1}\). From \(3\times4^{n-1}=768\), we get \(4^{n-1}=256=4^4\). Therefore, \(n-1=4\) and \(n=5\). At position 4, the term is 192, not 768. Exam tip: first identify the common ratio when finding a term's position.

Correct answer: C. 62

Explanation: The rule is to multiply the previous term by 2 and then add 2. Therefore, the term after 30 is \(2\times 30+2=62\). Although 60 is double of 30, it does not include the required addition of 2. Exam tip: In a recursive sequence, apply every part of the rule to the last given term.

Correct answer: B. 45

Explanation: The differences between consecutive terms are \(12-5=7\), \(21-12=9\), and \(32-21=11\). These are consecutive odd numbers, so the next difference is \(13\). Therefore, the next term is \(32+13=45\). Choosing \(47\) would give a difference of \(15\), which does not continue the pattern. Exam tip: For such sequences, first list the consecutive differences and check their pattern.

Correct answer: B. 61

Explanation: This is an arithmetic sequence with first term \(a=6\) and common difference \(d=5\). The \(n\)th term is \(a_n=a+(n-1)d\). Therefore, \(a_{12}=6+(12-1)\times5=6+55=61\). Hence, 61 is correct. Choosing 56 would count only 10 differences, whereas there are 11 differences from the first term to the 12th term. Exam tip: always use \((n-1)\), not \(n\), in the nth-term formula.

Correct answer: C. 34

Explanation: This is an arithmetic progression because each successive term decreases by 7. Here, \(a=90\), \(d=-7\), and \(n=9\). Thus, \(a_9=a+(9-1)d=90+8(-7)=34\). Therefore, 34 is correct. The value 36 would result from subtracting 7 only seven times, but there are eight gaps from the first term to the ninth term. Exam tip: for the \(n\)th term, always use \(n-1\) common differences.