Sequences

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 found by subtracting any term from the term immediately after it: 44 - 50 = -6. Checking the next pairs gives 38 - 44 = -6 and 32 - 38 = -6, so the difference is constant. Therefore the sequence is decreasing by 6 at every step, and the correct answer is option A, -6. Option C has the correct magnitude but the wrong sign; options B and D do not match the change between consecutive terms. A negative common difference correctly represents a decreasing arithmetic progression.

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.

In the sequence (5, 9, 13, 17), the differences between consecutive terms are 9−5=4, 13−9=4, and 17−13=4. Hence, its common difference is 4, so it is an arithmetic sequence. In (1, 3, 6, 10), the differences are 2, 3, and 4, which are not equal. Exam tip: subtract consecutive terms to check for a common difference.

In the sequence (18, 15, 12, 9), the differences between consecutive terms are 15−18 = −3, 12−15 = −3, and 9−12 = −3. Hence, 3 is subtracted each time. In option B, 4 is subtracted each time, so it is not correct. Exam tip: Subtract consecutive terms to check the common difference.

In option D, the terms are \(1^2, 2^2, 3^2, 4^2\), respectively, so it is a sequence of square numbers. Option A is a sequence of natural numbers, while B and C are multiples of 2 and 3 respectively. Exam tip: check whether each term can be written as \(n^2\).

Odd numbers are not exactly divisible by 2, such as 1, 3, 5, and 7. Therefore, (1, 3, 5, 7) is a sequence containing only odd numbers. In option A, every term is even because it is divisible by 2. Exam tip: A whole number ending in 1, 3, 5, 7, or 9 is odd.

Even numbers are divisible by 2 without a remainder. Every term in (2, 4, 6, 8) is divisible by 2, so it is a sequence of even numbers. In options B and D, some terms are even, but 3 and 5 are odd. Exam tip: Check every term of a sequence; having only some even terms is not enough.