Introduction to APs and common difference.

The common difference of an arithmetic progression is the difference between consecutive terms. Here, \(-7-(-12)=5\) and \(-2-(-7)=5\), so the common difference is 5. The value \(-5\) results from reversing the order of subtraction. Exam tip: always calculate common difference as second term minus first term.

In an arithmetic progression, the common difference is the difference between consecutive terms. Thus, second term − first term = 31 − 26 = 5. Therefore, 5 is correct. The number 26 is the first given term, not the common difference. Exam tip: When consecutive terms are given, subtract the earlier term from the later term.

The differences between consecutive terms are \(7-3=4\), \(12-7=5\), and \(18-12=6\). In an arithmetic progression, every consecutive difference must be the same, but here the differences are 4, 5, and 6. Therefore, this is not an arithmetic progression. Merely increasing terms do not make a sequence an AP. Exam tip: To check an AP, compare at least two or three consecutive differences.

Find the difference between two consecutive terms: \(19-12=7\). Also, \(26-19=7\) and \(33-26=7\), confirming that the difference is constant. Hence, each term is 7 greater than the previous term; this is the common difference of the AP. Choosing 6 does not match the actual difference between consecutive terms. Exam tip: In an AP, quickly find the common difference using \(d=a_2-a_1\).

In an arithmetic progression, the first term is \(a=3\) and the common difference is \(d=8\). So, the second term is \(3+8=11\), and the third term is \(11+8=19\). Hence, the first three terms are \((3, 11, 19)\). In option A, 8 has not been added to 3 to obtain the second term. Exam tip: check each successive term by adding \(d\).

An arithmetic progression is determined by its first term a and its common difference d. Starting with a = 4 means the first term must be 4. Adding d = 6 repeatedly gives the next terms: 4 + 6 = 10, 10 + 6 = 16, and 16 + 6 = 22. Therefore the sequence is (4, 10, 16, 22, …), so option A is correct. Option B begins with 6 and has difference 4, option C begins correctly but has difference 4, and option D has difference 6 but begins with 10. Checking both the first term and the repeated difference prevents selecting a partially matching sequence.

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 us to inspect only the four explicitly displayed terms, not to find every possible later term of the progression. Those four terms are 21, 24, 27, and 30. Hence 21, 24, and 30 are present, while 33 is absent from the displayed group. It is true that the common difference is 3 and 33 would be the next term after 30, but being a possible next term does not make it one of the four given terms. Therefore option D is the correct answer. The ellipsis indicates continuation, but the phrase “four given terms” limits the comparison to the first four listed terms.

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 of an arithmetic progression is found by subtracting one term from the next term. It tells us how much the sequence changes each time. A positive difference means the terms increase, while a negative difference means they decrease. In this sequence, the terms rise regularly, so the difference should be the same between every pair of neighboring terms.

The second term is 9 and the third term is 16. Therefore, the difference is calculated as \\(16-9=7\\). Checking the next pair also gives \\(23-16=7\\), confirming that the common difference is 7. Thus, choice C is correct. The wording should be understood as subtracting the second term from the third; reversing the order would give \\(-7\\), which is not the listed answer.

The difference of consecutive terms is (42-36=6). Always subtract the previous term from the next term to find the common difference.

For an arithmetic progression, subtract consecutive terms in the same order and check whether the result is always the stated common difference. In option A, 42 − 49 = −7, 35 − 42 = −7, and 28 − 35 = −7. Thus every step decreases by 7, exactly as required. Option B has common difference +7, option C has difference −9, and option D has difference +7. A negative common difference means the terms decrease as the sequence advances; it does not mean that the terms themselves must all be negative. Therefore option A is the only correct choice.