Sequences

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.

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.

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.

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: \(3,5,7,9,\ldots\).

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.

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.

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.

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.

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.

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.

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.

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 \(n\).

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.

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 \(n=k\).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.