Introduction to APs and common difference.

The common difference is the difference between consecutive terms. Here, the second term is −3 and the first term is −8, so \((-3)-(-8)=5\). Also, \(2-(-3)=5\) and \(7-2=5\), confirming the same difference. Therefore, the common difference is 5. The close distractor −5 results from reversing the subtraction or mishandling the negative sign. Exam tip: In an AP, calculate next term − previous term.

The common difference is found by subtracting a term from the term immediately after it. Using the first two terms, d=40−44=−4. The other consecutive differences confirm this: 36−40=−4 and 32−36=−4. The sequence decreases by 4 each time, so the difference must be negative. Therefore option B is correct; +4 would incorrectly ignore the direction of change.

The common difference of an arithmetic progression is the difference between consecutive terms: \(d=a_{n+1}-a_n\). Since each next term is 9 greater than the previous term, \(d=9\). Option D is incorrect because it represents decreasing terms. Exam tip: “that much more each time” gives a positive common difference, while “that much less” gives a negative one.

The common difference of an arithmetic progression is the difference between any term and the preceding term. Here, \(2.8-2.2=0.6\) and \(3.4-2.8=0.6\), so the common difference is \(0.6\). The option 0.4 does not represent the difference between consecutive terms. Exam tip: Subtract any term from the next term to find the common difference.

Compute the difference between consecutive fractional terms: d=5/3−2/3=(5−2)/3=3/3=1. Checking the next pair gives 8/3−5/3=3/3=1, so the difference is constant and the sequence is an arithmetic progression. Thus option B is correct. The numerator increases by 3 while the denominator remains 3, producing an increase of one whole unit, not one-third.

In an arithmetic progression, the difference between consecutive terms must remain constant. In option B, \(16-18=-2\), \(14-16=-2\), and \(12-14=-2\), so its common difference is \(d=-2\). Option A has a difference of \(+2\), the terms in option C are doubling, and option D has differences \(-2,-3,-4\). Exam tip: subtract each term from the next one and check whether the difference is constant.

The common difference of an arithmetic progression is the difference between two consecutive terms. Here, subtracting the first term from the second gives \((b+5)-b=5\). Therefore, the common difference is 5. The value 10 is the difference between the third and first terms, not the common difference. Exam tip: To find the common difference, subtract a term from the term immediately after it.

This is an arithmetic progression because the difference between consecutive terms is 7: \(14-7=7\), \(21-14=7\), and \(28-21=7\). Therefore, the next term is \(28+7=35\). Getting 34 would require adding 6, which is not the common difference of this sequence. Exam tip: To find the next term, first identify the difference between consecutive terms.

The terms of the given arithmetic progression are 3, 9, 15, and 21 in order. Thus, 3 is the first term, 9 is the second term, and 15 is the third term. Since 21 is the fourth term, it is not correct. Exam tip: always count the first listed term as the first term.

The defining property of an arithmetic progression is that the difference between every pair of consecutive terms is constant. If the terms are \(a_1,a_2,a_3,\ldots\), then \(a_2-a_1=a_3-a_2=a_4-a_3=d\), where \(d\) is the common difference. The terms can increase, decrease, or remain constant, so they need not all be positive.

For example, in \(3,7,11,15,\ldots\), each difference is 4, so it is an arithmetic progression. Equal ratios describe a geometric progression instead, not an arithmetic one. Likewise, squaring the previous term is not the defining rule, and negative or zero terms are allowed. Therefore the statement that consecutive terms have equal differences is correct, so choice B is the appropriate answer.

In an arithmetic progression, all consecutive differences are equal. Here the differences are not equal, so it is not an arithmetic progression.

In an AP, the second term is \(a_2=a+d\). Here, \(a_2=20+(-5)=20-5=15\). Therefore, option A is correct. 20 is the first term, while 25 would result if the common difference were \(+5\). Exam tip: Adding a negative common difference means subtracting its magnitude.

To find the common difference, subtract a term from the term immediately after it. Here, \(64-72=-8\) and \(56-64=-8\), so each successive term decreases by 8. Therefore, the common difference is \(-8\). The number \(8\) is only the magnitude of the decrease, not the common difference. Exam tip: use \(d=a_2-a_1\); if the terms decrease, \(d\) is negative.

The number of pages read on the first day is the first term, so \(a=5\). Since 3 more pages are read each day, the common difference is \(d=3\). Thus the terms are \(5, 5+3=8, 8+3=11, 11+3=14\). Option B has first term 3, so it is not correct. Exam tip: In an AP, first identify the first term and the fixed change between consecutive terms.

Term position is determined by counting from the first displayed term: 16 is the first term, 22 is the second, 28 is the third, and 34 is the fourth. Therefore the fourth term is 34, so option D is correct. Although the sequence has common difference 6 and could be handled by the nth-term formula, direct counting is the clearest method because the requested position is already shown.

In the sequence (1, 7, 13, 19, \ldots), the differences between consecutive terms are 7−1=6, 13−7=6, and 19−13=6. Therefore, its common difference is \(d=6\). Option C has common difference \(-6\), so it is not correct. 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 the fixed number added to one term to obtain the next term, so it is calculated as later term minus preceding term. Using the first two terms, d = (−4) − 0 = −4. The result can be checked with the next pairs: (−8) − (−4) = −4 and (−12) − (−8) = −4. Since the same value occurs each time, the sequence is an arithmetic progression with common difference −4. The negative sign is essential because the terms decrease by four units. Option B, 4, would indicate an increasing sequence, while option D is only one term and option A is not the actual change. Therefore option C is correct.

In an arithmetic progression, a denotes the first term and d denotes the constant difference between consecutive terms. The first displayed term is 13, so a = 13. The common difference is obtained by subtracting any term from the following term: d = 18 − 13 = 5. The check 23 − 18 = 5 and 28 − 23 = 5 confirms that the sequence is arithmetic with d = 5. Therefore option B is correct. Option A reverses the two quantities, option C treats the second term as the first term, and option D incorrectly uses the second term as the difference. The word respectively requires the answers in the order a, then d.

In an arithmetic progression, the common difference d is the difference between two consecutive terms. When d=0, there is no change from one term to the next, so all consecutive terms are equal. In options B and D, the common differences would be 1 and -1 respectively, so they are not correct. Exam tip: An AP with d=0 is a constant sequence, for example 5, 5, 5, ... .

This is an arithmetic progression because the difference between consecutive terms is 4. The fourth term is 18, so the fifth term is 18+4=22. The value 24 would require a common difference of 6, which is not the case here. Exam tip: To find the next term, add the common difference to the previous term.

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 \(d\).