Sequences

Correct answer: B. 1

Explanation: Each term of the first sequence increases by 3, so its 6th term is \(2+5\times3=17\). The 6th term of the second sequence is \(1+5\times3=16\). Therefore, the difference is \(17-16=1\). The nearby option 0 is incorrect because corresponding terms of the two sequences differ by 1. Exam tip: for two sequences with the same common difference, the difference between corresponding terms remains constant.

Correct answer: A. The first is greater by (6)

Explanation: Both sequences have difference (5), so the term difference always remains (6). The (8)th terms also differ by (6).

Correct answer: C. 65

Explanation: This is an arithmetic sequence with first term 3 and common difference 5. Its first 5 terms are 3, 8, 13, 18, and 23. Therefore, \(3+8+13+18+23=65\). A result such as 60 can arise from adding a term incorrectly. Exam tip: write the required terms first, then check the addition.

Correct answer: B. 136

Explanation: The first four terms are 40, 36, 32, and 28 because each successive term decreases by 4. Therefore, \(40+36+32+28=136\). A result of 140 can arise from calculating the fourth term incorrectly. Exam tip: Write all the required terms before adding them.

Correct answer: B. 8th term

Explanation: This is an arithmetic progression with first term 1 and common difference 3. Its nth term is \(a_n=1+(n-1)\times3\). Putting \(a_n=22\), we get \(1+(n-1)\times3=22\), so \(n=8\). Hence, 22 is the 8th term. The 9th term is 25, so it cannot be correct. Exam tip: To find the position of a given term, substitute its value in \(a_n=a+(n-1)d\).

Correct answer: A. 43

Explanation: This is an arithmetic progression with first term 5 and common difference 3. Hence, the sixth term is 20 and the seventh term is 23. Therefore, \(a_6+a_7=20+23=43\). Choosing 45 would result from an error in finding or adding the terms. Exam tip: Write the terms in order first and check their positions carefully.

Correct answer: B. 18

Explanation: Each successive term in the sequence increases by 6. Thus, the second term is \(a_2=18\) and the fifth term is \(a_5=36\). Therefore, \(a_5-a_2=36-18=18\). Option 24 is the fourth term, not the difference between the required terms. Exam tip: identify the required term positions first, then subtract their values.

Correct answer: C. \(12\)

Explanation: In the sequence, the first term is \(a_1=9\) and the fourth term is \(a_4=21\). Therefore, \(a_4-a_1=21-9=12\), so \(12\) is correct. Choosing \(14\) would give an incorrect difference between the terms. Exam tip: subtract the lower-indexed term from the higher-indexed term when finding \(a_n-a_m\).

Correct answer: C. 26

Explanation: This is an arithmetic progression with first term 6 and common difference 4. To reach the sixth term, 4 is added 5 times: \(a_6=6+(6-1)\times4=26\). Therefore, 26 is correct. Getting 24 would mean adding the difference only 4 times. Exam tip: use \(a_n=a_1+(n-1)d\) to find the \(n\)th term.

Correct answer: C. \(5, 10, 15, 20\)

Explanation: Option C ends at \(20\), so it has exactly four terms and is a finite sequence. In options A, B, and D, \(\ldots\) indicates that the terms continue, so they are infinite sequences. Exam tip: a sequence with a definite last term is finite.

Correct answer: C. 47

Explanation: The rule is: each term equals twice the previous term plus 1. Hence, the term after 23 is \(2\times 23+1=46+1=47\). Option 46 is only double of 23; it misses the additional 1. In exams, apply the stated rule directly to the last given term.

Correct answer: B. 4.5

Explanation: The rule is to divide the previous term by 2 and then subtract 1. Therefore, the term after 11 is \(11\div 2-1=5.5-1=4.5\). The value 5.5 is obtained only after dividing 11 by 2; subtracting 1 is still required. Exam tip: In recursive sequences, follow the stated order of operations carefully.

Correct answer: B. 34

Explanation: The differences between consecutive terms are 6−4=2, 10−6=4, 16−10=6, and 24−16=8. These differences increase by 2 each time, so the next difference is 10. Therefore, the next term is 24+10=34. Choosing 32 would repeat a difference of 8, whereas the difference pattern should be 2, 4, 6, 8, 10. Exam tip: For such sequences, first list consecutive differences and look for their pattern.

Correct answer: B. 45

Explanation: The differences between consecutive terms are 3, 6, 9, and 12. Since the amount subtracted increases by 3 each time, the next subtraction is 15. Therefore, \(60-15=45\). Option 42 would require subtracting 18, which does not follow the pattern. Exam tip: Write the differences between consecutive terms first to identify the pattern.

Correct answer: C. (120)

Explanation: The sequence follows a multiplying pattern rather than a constant-addition pattern. Starting with 1, multiply by 2 to get 2. Then multiply by 3 to get 6, and multiply by 4 to get 24. The multipliers increase successively as 2, 3, and 4, so the next multiplier is 5.

Multiplying the last known term by this next factor gives \\(24\times5=120\\). Therefore the next term is 120, which is option C. Another way to view the sequence is through factorials: \\(1=1!\\), \\(2=2!\\), \\(6=3!\\), and \\(24=4!\\), so the next term is \\(5!=120\\). Choices such as 96 do not continue the observed multiplication pattern.

Correct answer: A. \(2:5\)

Explanation: The fourth term of the first sequence is \(54\), and that of the second sequence is \(135\). Therefore, the ratio is \(54:135\). Dividing both terms by \(27\) gives \(54:135=2:5\). \(5:2\) is the reverse of the required ratio. Exam tip: Keep the order of terms in a ratio the same as the order of the sequences in the question.

Correct answer: B. 5

Explanation: This is an arithmetic progression with first term 6 and common difference 4. Its first 5 terms are 6, 10, 14, 18, and 22, and their sum is 6 + 10 + 14 + 18 + 22 = 70. Therefore, the correct answer is 5. The sum of the first 4 terms is only 48. Exam tip: use \(S_n=\frac{n}{2}[2a+(n-1)d]\) for the sum of the first n terms of an arithmetic progression.

Correct answer: C. 29

Explanation: Each term in this sequence is obtained by adding 5 to the previous term: 4, 9, 14, 19, 24, 29, 34. Therefore, the term immediately before 34 is 29. Option 31 is not a term of this sequence. Exam tip: Identify the common difference and move backward from the given term when asked for the preceding term.

Correct answer: B. 6

Explanation: Each term in the sequence is half of the preceding term: 96, 48, 24, 12, 6, 3. Therefore, the term immediately before 3 is 6. Although 12 occurs earlier in the sequence, 6 comes between 12 and 3, so 12 is not the immediate preceding term. Exam tip: identify the pattern and extend the sequence to find a required next or previous term.

Correct answer: A. \(3n+4\)

Explanation: This is an arithmetic progression with first term \(a=7\) and common difference \(d=3\). Therefore, \(a_n=a+(n-1)d=7+(n-1)\times3=3n+4\). The close distractor \(3n+7\) gives \(10\) when \(n=1\), so it cannot represent this sequence. Exam tip: use \(a_n=a+(n-1)d\) after identifying the first term and common difference.

Correct answer: A. \(a_n=55-5n\)

Explanation: This is an arithmetic sequence with first term \(a_1=50\) and common difference \(d=-5\). Therefore, \(a_n=a_1+(n-1)d=50+(n-1)(-5)=55-5n\). In option A, \(n=1\) gives 50 and \(n=2\) gives 45. Option B gives 45 when \(n=1\), so it is incorrect. Exam tip: In \(a_n=a_1+(n-1)d\), do not forget the \(n-1\) term.

Correct answer: C. 30

Explanation: The rule says that each new term equals the sum of the two immediately preceding terms. Thus, \(a_3=3+8=11\), \(a_4=8+11=19\), and \(a_5=11+19=30\). Therefore, the correct answer is 30. Note that 19 is the fourth term, not the fifth. Exam tip: In a recursive sequence, write the terms in order and check their subscripts carefully.

Correct answer: B. 5

Explanation: Each term is obtained by multiplying the previous term by 3: 2, 6, 18, 54, 162. Therefore, 162 is the fifth term, so \(n=5\). Since 54 is the fourth term, 4 is a close but incorrect option. Exam tip: identify the multiplication pattern before counting the terms.

Correct answer: C. 52

Explanation: This is an arithmetic sequence because 6 is added to each successive term. Here, the first term is \(a=4\) and the common difference is \(d=6\). The \(n\)th term is \(a_n=a+(n-1)d\). Therefore, \(a_9=4+(9-1)\times6=4+48=52\). Hence, 52 is the correct answer. A common error leading to 50 is counting the gaps between terms incorrectly. Exam tip: from the first term to the \(n\)th term, there are \(n-1\) common differences.

Correct answer: A. 28

Explanation: This is an arithmetic sequence because each successive term decreases by 6. Here, the first term is \(a=64\) and the common difference is \(d=-6\). Therefore, \(a_7=a+(7-1)d=64+6(-6)=28\). Choosing 30 usually results from counting the number of subtractions incorrectly. Exam tip: for the \(n\)th term, use \(a_n=a+(n-1)d\).