Sequences

Correct answer: A. 16

Explanation: Each term in the sequence is 3 greater than the previous term: 7−4=3, 10−7=3, and 13−10=3. Therefore, the next term is 13+3=16. Choosing 17 would require adding 4 to 13, which does not follow the pattern. Exam tip: Find the difference between consecutive terms to identify a sequence pattern.

Correct answer: B. 40

Explanation: Each consecutive term decreases by 5: from 60 to 55, 55 to 50, and 50 to 45. Therefore, the next term is \(45-5=40\). While 35 would be the term after that, 45 and 50 are already given terms. Exam tip: To find the next term, first identify the common difference between consecutive terms.

Correct answer: C. 243

Explanation: In this sequence, each term is 3 times the preceding term: \(3\to9\to27\to81\). Therefore, the next term is \(81\times3=243\). \(216\) is incorrect because it is not 3 times 81. Exam tip: first check whether consecutive terms follow a multiplication or division pattern.

Correct answer: D. 17

Explanation: The difference between consecutive terms is 4: 5−1=4, 9−5=4, and 13−9=4. Therefore, the term after 13 is 13+4=17. Choosing 16 would give a difference of 3, so it is incorrect. Exam tip: check the difference between consecutive terms before finding the next term.

Correct answer: A. 24

Explanation: In this sequence, each term is 6 more than the previous term: 6, 12, 18, 24, 30. Therefore, the term after 18 is 24. Choosing 22 or 20 would not maintain the common difference of 6. Exam tip: For a missing-term sequence, first check the difference between consecutive known terms.

Correct answer: B. 80

Explanation: In this sequence, each next term is 10 less than the previous term: 100, 90, 80, 70, 60. Therefore, the term after 90 and before 70 is 80. Choosing 75 would not keep the difference between consecutive terms constant. Exam tip: In a missing-term sequence, compare consecutive terms to identify the rule.

Correct answer: C. 20

Explanation: The rule for the terms is \(a_n=4n\). To find the fifth term, substitute \(n=5\): \(a_5=4\times 5=20\). Hence, 20 is correct. The value 16 comes from \(4\times4\), which is the fourth term \(a_4\), not the fifth term. Exam tip: the subscript in \(a_n\) shows the term position, so substitute that number for \(n\).

Correct answer: D. 10

Explanation: Given \(a_n=n^2+1\), substitute \(n=3\): \(a_3=3^2+1=9+1=10\). Therefore, 10 is correct. Option 9 is only the value of \(3^2\); it misses the added \(+1\). Exam tip: To find a term of a sequence, first substitute the term number for \(n\), then simplify carefully.

Correct answer: B. \(-6\)

Explanation: To find the common difference, subtract a term from the term immediately after it: \(36-42=-6\). Similarly, \(30-36=-6\), so the common difference between consecutive terms is \(-6\). Although the terms decrease by 6, the difference itself must be negative. Exam tip: always calculate common difference as second term minus first term.

Correct answer: C. By multiplying by 3

Explanation: To get 6 from 2, 18 from 6, and 54 from 18, the preceding term is multiplied by 3 each time: \(2\times3=6\), \(6\times3=18\), and \(18\times3=54\). Therefore, multiplying by 3 is correct. Adding 6 works only for the first pair, since \(6+6=12\), not 18. Exam tip: check a sequence rule using at least two consecutive pairs of terms.

Correct answer: D. By dividing by 2

Explanation: In the sequence, 80 becomes 40, 40 becomes 20, and 20 becomes 10 by dividing the preceding term by 2 each time. Therefore, the correct rule is division by 2. Subtracting 10 would give 70 after 80, while dividing by 4 would give 20 after 80. Exam tip: Compare consecutive terms using their ratio to quickly identify a multiplication or division pattern.

Correct answer: A. 36

Explanation: The terms are squares of consecutive natural numbers: \(1^2, 2^2, 3^2, 4^2, 5^2\). Therefore, the next term is \(6^2=36\). Although 49 is also a square number, it is \(7^2\) and comes after 36. Exam tip: check consecutive differences in such sequences; here they are \(3,5,7,9\), so the next difference is 11.

Correct answer: B. 216

Explanation: The terms are consecutive cubes of natural numbers: \(1=1^3\), \(8=2^3\), \(27=3^3\), \(64=4^3\), and \(125=5^3\). Therefore, the next term is \(6^3=216\). Note that \(225\) is not a perfect cube. Exam tip: Rewrite sequence terms as \(1^3, 2^3, 3^3\), and so on, to identify the pattern quickly.

Correct answer: C. 22

Explanation: The differences between consecutive terms are 2, 3, 4, and 5. Since each difference increases by 1, the next difference is 6. Therefore, the next term is \(16+6=22\). Choosing 21 would give a difference of only 5, whereas the differences should follow 2, 3, 4, 5, 6. Exam tip: For such sequences, first write the consecutive differences and identify their pattern.

Correct answer: D. 15

Explanation: The successive terms are obtained by subtracting 1, 2, 3, and 4: 30 to 29, 29 to 27, 27 to 24, and 24 to 20. Therefore, the next subtraction is 5: \(20-5=15\). Hence, 15 is the correct answer. Option 16 would result from subtracting 4 again, but the pattern of subtractions is 1, 2, 3, 4, 5. Exam tip: For such sequences, write the differences between consecutive terms and look for their pattern.

Correct answer: A. 13

Explanation: In this sequence, each term is the sum of the two immediately preceding terms: \(1+1=2\), \(1+2=3\), \(2+3=5\), and \(3+5=8\). Therefore, the next term is \(5+8=13\). Although 12 may seem like a simple continuation after 8, it does not follow the sequence rule. Exam tip: For such sequences, check whether adding consecutive previous terms produces the next term.

Correct answer: B. 37

Explanation: From the third term onward, each term is the sum of the two immediately preceding terms: \(9=4+5\), \(14=5+9\), and \(23=9+14\). Therefore, the next term is \(14+23=37\). Option 36 would require adding 14 and 22, which does not follow the pattern. Exam tip: For sequence questions, first check successive differences and then check sums of adjacent terms.

Correct answer: A. (32, 28, 24, 20)

Explanation: In the sequence (32, 28, 24, 20), the differences between consecutive terms are 28 − 32 = −4, 24 − 28 = −4, and 20 − 24 = −4. Hence, 4 is subtracted each time. In option B, 6 is subtracted each time, so it is not correct. Exam tip: Find the difference between consecutive terms to check the rule of a sequence.

Correct answer: B. (6,12,18,24)

Explanation: (6,12,18,24) are all multiples of (6). In a multiples sequence, the first term and difference both help.

Correct answer: C. (1, 8, 27, 64)

Explanation: In option C, the terms are \(1^3=1\), \(2^3=8\), \(3^3=27\), and \(4^3=64\). Therefore, it is a sequence of cube numbers. Option A is a sequence of square numbers, since its terms are \(1^2, 2^2, 3^2, 4^2\). Exam tip: find a cube by multiplying a number by itself three times; for example, \(3^3=3\times3\times3=27\).

Correct answer: A. \(14\)

Explanation: To find the fourth term, substitute \(n=4\) in \(a_n=3n+2\): \(a_4=3\times4+2=12+2=14\). Hence, \(14\) is correct. \(12\) is only the value of \(3\times4\); the additional \(2\) must also be included. Exam tip: For an \(n\)th-term question, first substitute the given value of \(n\) correctly into the rule.

Correct answer: B. The statement is incorrect; 15 is the fifth term.

Explanation: Count the terms in order: 3 is first, 6 second, 9 third, 12 fourth, and 15 fifth. Hence Reena has stated the position incorrectly. Exam tip: always begin numbering from the first listed term.

Correct answer: C. 7

Explanation: Given (a_n=12-n), substitute (n=5) to find the fifth term: (a_5=12-5=7). Therefore, the correct answer is 7. Getting 6 may result from incorrect subtraction or using the wrong term number. Exam tip: for an nth-term formula, substitute the given value of (n) carefully first.

Correct answer: C. 3

Explanation: To find the first term, substitute \(n=1\) in the given formula. Thus, \(a_1=5(1)-2=5-2=3\). Therefore, the correct answer is 3. The value 2 may result from an incorrect calculation of \(5-2\). Exam tip: for the \(k\)th term of a sequence, always put \(n=k\) in the formula.

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

Explanation: Substituting \(n=1,2,3,4\) in the rule gives \(a_1=4(1)-1=3\), \(a_2=7\), \(a_3=11\), and \(a_4=15\). Therefore, the first four terms are \((3,7,11,15)\). In option B, the subtraction of 1 has been ignored. Exam tip: for the first term of a sequence, usually begin with \(n=1\), not \(n=0\).