Introduction to APs and common difference.

An arithmetic progression is identified by its first term a and its constant common difference d. The required sequence must begin with 4, and every following term must be 6 greater than the previous one. Starting from 4 gives 4 + 6 = 10, 10 + 6 = 16, and 16 + 6 = 22, so the sequence is (4, 10, 16, 22, …). Hence option A is correct. Option B has first term 6 and common difference 4. Option C begins with 4 but its difference is 4 rather than 6. Option D has difference 6, but its first term is 10. Thus both defining conditions, not merely one of them, must be checked before selecting the progression.

Each term in this sequence is 10 less than the preceding term, so the common difference is \(d=-10\). After 60, we get \(60-10=50\), and then \(50-10=40\). Hence, the next two terms are (50, 40). (55, 50) is incorrect because it uses a difference of -5. Exam tip: first check the difference between consecutive terms before finding later terms.

The wording asks which term is absent from the four terms explicitly displayed. Those four given terms are 21, 24, 27, and 30. Therefore 21, 24, and 30 are present, while 33 is not one of the four written terms, so option D is correct. The common difference is 3, and consequently 33 would be the next term after 30. However, the ellipsis only indicates that the progression continues; it does not make 33 part of the four terms already listed. This distinction between the displayed initial terms and a later term is important. The other three choices can be directly located in the given list.

The difference between consecutive terms is 12 - 12 = 0, and this difference is the same for every pair of consecutive terms. Hence, it is an arithmetic progression with common difference d = 0 and constant terms. An increasing or decreasing AP has a positive or negative common difference, respectively. Exam tip: To identify the type of an AP, check the sign of its common difference d.

To find the common difference, subtract consecutive terms: 11 - 8 = 3 and 14 - 11 = 3. Since each successive row has 3 more bricks, the common difference is 3. Here, 8 is the first term, not the common difference. Exam tip: In an AP, common difference = second term − first term.

From 35 to 30 and from 30 to 25, 5 is subtracted each time. Therefore, each term is 5 less than the preceding term. The common difference is \(d=-5\); the negative sign shows decrease, whereas the question asks how much less, so the answer is 5. Exam tip: In a decreasing AP, the common difference is negative.

In an arithmetic progression, the common difference is the difference between consecutive terms. Here, \(\frac{3}{5}-\frac{1}{5}=\frac{2}{5}\). Also, \(1-\frac{3}{5}=\frac{2}{5}\), confirming that the common difference is \(\frac{2}{5}\). \(\frac{1}{5}\) is the first term, not the common difference. Exam tip: calculate common difference as next term minus previous term.

In an arithmetic progression, the common difference is \(d=a_2-a_1\). Hence, \(d=18-25=-7\). Therefore, \(-7\) is correct. Choosing \(7\) would miss the negative sign, since the second term is 7 less than the first term. Exam tip: always calculate \(d\) as next term − previous term.

The consecutive differences are \(22-11=11\), \(33-22=11\), and \(44-33=11\). Since the difference between every pair of consecutive terms is constant, the sequence is an arithmetic progression. A geometric progression requires a constant ratio between consecutive terms, which is not the case here. Exam tip: To identify an AP, check whether two or three consecutive differences are equal.

The position of a term is found by counting from the beginning of the sequence, with the first term assigned position 1. In an arithmetic progression, the term number can also be checked using \(a_n=a+(n-1)d\), but direct counting is simplest when the required term is visibly listed. The value of a term and its position are different ideas.

In \(18,23,28,33,\ldots\), the order is: 18 is term 1, 23 is term 2, 28 is term 3, and 33 is term 4. Therefore 23 is the second term, so choice B is correct. The common difference is 5, but that does not affect the requested position. Calling 23 the first term would ignore the term 18 that comes before it.

The common difference of an AP is the difference between two consecutive terms: \(d=a_2-a_1=48-54=-6\). Thus, each next term is 6 less than the previous term, so \(-6\) is correct. \(6\) gives only the magnitude of the difference, not its correct sign. Exam tip: always calculate common difference as second term minus first term.

In an arithmetic progression, the common difference is d = second term − first term. Therefore, d = 14 − 10 = 4. Here, 10 is the first term and 14 is the second term, so neither of them is the common difference. Exam tip: To find d in an AP, subtract the preceding term from the next term.

The common difference is the difference between consecutive terms:
\[d=9-2=7.\]
Also, \(16-9=7\) and \(23-16=7\), so \(d\) is consistent.

The direct answer is option C, 6. The common difference is the difference between consecutive terms, calculated as next term minus previous term. Using the first two terms, \(d=42-36=6\). Check the next pairs: \(48-42=6\) and \(54-48=6\), so the difference is constant and the answer is confirmed. Option A, 4, is not the gap between 36 and 42. Option B, 5, is also not obtained from subtraction. Option C, 6, matches every consecutive pair and is correct. Option D, 12, is twice the actual difference and may result from an incorrect operation or combining gaps. Memory cue: in an AP, subtract neighbouring terms and verify with another pair.

To identify the required arithmetic progression, subtract each term from the following term and check whether every result is −7. In option A, 42 − 49 = −7, 35 − 42 = −7, and 28 − 35 = −7. Thus the terms decrease by exactly 7 at every step, so option A satisfies the condition. Option B has common difference +7, option C has common difference −9 because 40 − 49 = −9, and option D has common difference +7. A negative common difference means that the sequence decreases as it proceeds; it does not require the terms themselves to be negative. Therefore option A is the only correct answer.

In an arithmetic progression, the first term is a and the common difference d is added to obtain each next term. Here, a = 11 and d = 6: 11, 11 + 6 = 17, 17 + 6 = 23, 23 + 6 = 29. Hence, the first four terms are (11, 17, 23, 29). In option A, 5 has been added instead of 6. Exam tip: Check that the difference between every pair of consecutive terms equals d.

Check the difference between consecutive terms: \(75-81=-6\), \(69-75=-6\), and \(63-69=-6\). Thus, this is an arithmetic progression with common difference \(-6\). Therefore, the next term is \(63-6=57\). Getting 60 would require subtracting 3, which does not follow the given common difference. Exam tip: For a next-term question, first find the difference between consecutive terms and apply it to the last term.

In an AP, the common difference is the difference between consecutive terms. \( d=\frac{6}{7}-\frac{4}{7}=\frac{2}{7} \). Checking next: \( \frac{8}{7}-\frac{6}{7}=\frac{2}{7} \) also matches, so \( d=\frac{2}{7} \).

The common difference of an arithmetic progression is obtained by subtracting the earlier term from the immediately following term. Since the consecutive terms are 32 and 41 in that order, d = 41 − 32 = 9. Therefore the progression increases by 9 at this step, and option C is correct. The value 73 comes from adding the two terms rather than finding their difference. Values 7 and 8 result from incorrect subtraction or estimation. No nth-term or sum formula is needed because two consecutive terms directly reveal the common difference. The same d would occur between every neighboring pair in a true arithmetic progression.

Answer: 9, so option D is correct. In an arithmetic progression, the difference between every two consecutive terms is the same. The known consecutive terms 14 and 19 allow us to find the common difference: d = 19 - 14 = 5. The term immediately before 14 must therefore be 5 less than 14, because adding 5 to it must produce 14. Hence k = 14 - 5 = 9. We can verify the complete beginning of the progression: 4, 9, 14, 19, ... . Its consecutive differences are 9 - 4 = 5, 14 - 9 = 5, and 19 - 14 = 5, so the defining condition is satisfied. Option A, 6, would give differences 2 and 8 around k; option B, 7, would give 3 and 7; and option C, 8, would give 4 and 6. None of these has equal consecutive differences. A useful memory cue is: in an AP, move backward by subtracting the common difference and move forward by adding it.

The governing concept is that every pair of consecutive terms in an arithmetic progression has the same common difference. Using the known terms, the difference is d = 15 − 11 = 4. This is confirmed by 19 − 15 = 4, so the progression increases by 4 at every step. The term immediately before 11 must therefore be found by moving one step backward and subtracting 4: p = 11 − 4 = 7. Hence option C is correct. Option A would make the difference from p to 11 equal to 8, option B would make it 6, and option D would make it 2. None of those values preserves the constant difference of 4 across the sequence.

Answer: 9, 12, 15, 18, ..., so option B is correct. In the notation of an arithmetic progression, a is the first term and d is the common difference between consecutive terms. We must check both conditions, not just one. Option B begins with 9, so its first term is a = 9. Its differences are 12 - 9 = 3, 15 - 12 = 3, and 18 - 15 = 3. Therefore it has both the required first term and the required common difference. Option A has difference 3, but it starts with 3 rather than 9. Option C starts with 9, but its difference is 2, not 3. Option D has difference 3, but its first term is 12. Thus only B satisfies the complete requirement. A common confusion is to identify a sequence merely because its difference is correct; always verify the first term too.

Number the terms from left to right starting with position 1: a₁=14, a₂=20, a₃=26, and a₄=32. Therefore 20 is the second term, so option B is correct. The common difference is 6, but it is not necessary to calculate it for this question. The first term is 14, and 26 and 32 occupy the third and fourth positions respectively.

The relevant concept is the common difference of an arithmetic progression. Compute each requested difference directly: the difference of the first two terms is 12 - 5 = 7, while the difference of the third and fourth terms is 26 - 19 = 7. Since both results are equal, the sequence is consistent with an arithmetic progression having d = 7. Option A reports both calculated differences. Option B incorrectly uses 5, which is the first term rather than a difference; option C lists terms instead of subtracting them; and option D doubles the second difference without justification. Hence option A is correct.

The position of a term is counted from the beginning of the sequence. Here, 25 is the first term, 21 follows it, so 21 is the second term. The common difference is -4, confirming that the displayed order is 25, 21, 17, 13. Therefore, option B is correct; options A, C and D identify the positions of 25, 17 and 13 respectively.