Sequences

Correct answer: A. 15

Explanation: The difference between consecutive terms is 3: 6−3=3, 9−6=3, and 12−9=3. Therefore, adding 3 to 12 gives the next term, 15. The number 18 would come after adding 3 twice from 12. Exam tip: Check the difference between consecutive terms before finding the next term.

Correct answer: B. 13

Explanation: Check the difference between consecutive terms: 22 - 25 = -3, 19 - 22 = -3, and 16 - 19 = -3. Thus, each next term is obtained by subtracting 3. Therefore, 16 - 3 = 13, so 13 is correct. Choosing 14 would mean a decrease of only 2. Exam tip: Before finding the next term, check the difference between consecutive terms.

Correct answer: C. 32

Explanation: In this sequence, each term is twice the previous term: 2, 4, 8, 16. Therefore, the next term is \(16\times 2=32\). Although 24 is obtained by adding 8 to 16, the differences are 2, 4, and 8, so they are not constant. Exam tip: first check whether consecutive terms are related by multiplication or division.

Correct answer: D. 25

Explanation: The given terms are consecutive square numbers: \(1=1^2\), \(4=2^2\), \(9=3^2\), and \(16=4^2\). Therefore, the next term is \(5^2=25\). \(24\) is not correct because it is not a perfect square. Exam tip: In a sequence of square numbers, the successive differences are odd numbers: \(3,5,7,9,\ldots\).

Correct answer: A. 30

Explanation: The differences between consecutive terms are 6−2=4, 12−6=6, and 20−12=8. Since each difference increases by 2, the next difference is 10. Therefore, the next term is 20+10=30. Choosing 28 would give a difference of only 8, which does not continue the pattern of increasing differences. Exam tip: For such sequences, first list the consecutive differences and check their pattern.

Correct answer: B. 80

Explanation: In this sequence, each term is twice the preceding term: 5 to 10, 10 to 20, and 20 to 40. Therefore, the next term is \(40 \times 2 = 80\). Although 60 is obtained by adding 20 to 40, the differences are not constant here; the pattern is multiplication by 2. Exam tip: For a sequence, first check consecutive differences and then check consecutive ratios.

Correct answer: C. 4

Explanation: In this sequence, each term is obtained by dividing the previous term by 2: 64, 32, 16, 8. Therefore, the next term is \(8 \div 2=4\). The number 2 would be obtained after halving 8 twice, so it is not the next term. Exam tip: First check for a common multiplication or division rule between consecutive terms.

Correct answer: D. 19

Explanation: Each term in the sequence is 3 greater than the previous term: 10, 13, 16, 19, 22. Therefore, the missing term after 16 is 19. Choosing 18 would not maintain the common difference of 3. Exam tip: For a missing-term sequence, first check the difference between consecutive known terms.

Correct answer: A. 21

Explanation: Each term in the sequence is obtained by adding 7: 7, 14, 21, 28, 35. Therefore, the missing term is 21. If 20 were used, the increase from 14 to 20 would be 6, so the common difference would not remain the same. Exam tip: For a missing-term sequence, compare consecutive known terms to identify the rule.

Correct answer: B. 35

Explanation: Each successive term is 5 less than the preceding term: 40, 35, 30, 25, 20. Therefore, the missing term after 40 is 35. If 34 were chosen, the difference from 40 would be 6, so the sequence rule would not be followed. Exam tip: Compare consecutive known terms to identify the common difference in a sequence.

Correct answer: B. Reena is incorrect; 18 is not a term of this sequence.

Explanation: The common difference is 3. After 13, the next terms are 16 and 19, so 18 does not occur in the sequence. Exam tip: check the common difference first.

Correct answer: D. 8

Explanation: The rule for the terms is \(a_n=n+5\). For the third term, substitute \(n=3\): \(a_3=3+5=8\). Therefore, the correct answer is 8. Getting 9 would mean using \(n=4\), which gives the fourth term \(a_4\), not the third term. Exam tip: In \(a_n\), substitute the required term number for \(n\).

Correct answer: A. 14

Explanation: To find the fifth term, substitute n=5 in the rule: a_5=3(5)-1=15-1=14. Therefore, the correct answer is 14. The value 15 is only 3×5; 1 still has to be subtracted. Exam tip: In an nth-term rule, replace n with the required term number before calculating.

Correct answer: B. \(6\)

Explanation: To find the first term, substitute \(n=1\) in the formula: \(a_1=4(1)+2=6\). Therefore, the correct answer is \(6\). The number \(4\) is only the coefficient of \(n\), not the first term. Exam tip: To find the \(k\)th term from a formula, put \(n=k\).

Correct answer: C. 5

Explanation: Find the difference between consecutive terms: \(8-3=5\), \(13-8=5\), and \(18-13=5\). Since the difference is 5 each time, the common difference is 5. Although 4 may seem plausible, it is not obtained by subtracting any consecutive terms here. Exam tip: subtract the first term from the second term to find the common difference.

Correct answer: A. -6

Explanation: The governing concept is the common difference of an arithmetic progression. It is calculated by subtracting one term from the next term. Here, 44 - 50 = -6. The result is confirmed by the following pairs: 38 - 44 = -6 and 32 - 38 = -6. Since the difference is constant, the sequence is an arithmetic progression with common difference -6. Therefore option A is correct. The negative sign is important because the terms decrease by 6 at each step. Option C has the correct magnitude but the wrong sign, while options B and D do not equal the change between any two consecutive terms. A common difference need not be positive; a negative value simply indicates a decreasing sequence.

Correct answer: B. (3, 5, 7, \ldots)

Explanation: Subtract each term from the next term: \(5-2=3\), \(10-5=5\), and \(17-10=7\). Hence, the sequence of consecutive differences is \((3, 5, 7, \ldots)\). Option A is incorrect because the actual difference between the first two terms is 3, not 2. Exam tip: To find consecutive differences, subtract each term from the term immediately following it.

Correct answer: C. 8

Explanation: In this sequence, each term is the sum of the two terms immediately before it: \(1+1=2\), \(1+2=3\), and \(2+3=5\). Therefore, the next term is \(3+5=8\). Option 7 is incorrect because it is not the sum of the previous two terms. Exam tip: For a sequence, first check patterns involving addition, subtraction, multiplication, or division between consecutive terms.

Correct answer: D. 21

Explanation: In this sequence, each term is the sum of the two preceding terms: \(2+3=5\), \(3+5=8\), and \(5+8=13\). Therefore, the next term is \(8+13=21\). An option such as 20 is incorrect because it is not the sum of the previous two terms. Exam tip: For such sequences, check differences between consecutive terms or sums of preceding terms.

Correct answer: A. 16

Explanation: The differences between consecutive terms are 1, 2, 3, and 4: 2-1=1, 4-2=2, 7-4=3, and 11-7=4. Therefore, the next difference is 5. So the next term is 11+5=16. Choosing 15 would give a difference of only 4, but the differences are increasing in order. Exam tip: For such sequences, first write the consecutive differences to identify the pattern.

Correct answer: B. 5

Explanation: The sequence subtracts 1, 2, 3 and 4 successively: \(20-1=19\), \(19-2=17\), \(17-3=14\), and \(14-4=10\). Therefore, subtract 5 next: \(10-5=5\). Hence, the correct answer is 5. Getting 6 would mean subtracting 4 again, but the number being subtracted increases by 1 each time. Exam tip: Write the consecutive differences to identify the rule in such sequences quickly.

Correct answer: C. 125

Explanation: The terms are consecutive cubes: \(1=1^3\), \(8=2^3\), \(27=3^3\), and \(64=4^3\). Therefore, the next term is \(5^3=125\). \(121\) is a perfect square, \(11^2\), so it cannot be the next term in this cube sequence. Exam tip: To identify a pattern, first check whether the terms are squares, cubes, or other powers.

Correct answer: D. 15

Explanation: Look at the differences between consecutive terms: \(3-1=2\), \(6-3=3\), and \(10-6=4\). The difference increases by 1 each time, so the next difference is \(5\). Therefore, the next term is \(10+5=15\). Choosing 14 would incorrectly keep the next difference as 4. Exam tip: For number sequences, first write the consecutive differences and identify their pattern.

Correct answer: A. By multiplying by 2

Explanation: In the sequence, 3 becomes 6, 6 becomes 12, and 12 becomes 24 by multiplying the previous term by 2 each time. Therefore, multiplying by 2 is correct. Adding 6 appears to work from 6 to 12, but it does not work from 3 to 6. In exams, check the rule using at least two consecutive pairs of terms.

Correct answer: B. By dividing by 2

Explanation: Dividing 36 by 2 gives 18, and dividing 18 by 2 gives 9. Therefore, each term is obtained by dividing the previous term by 2. Subtracting 9 would give 27 from 36, so that is not the rule. Exam tip: For successive terms of a sequence, first check for a multiplication or division relationship.