Introduction to APs and common difference.

Correct answer: B. (-4), negative

Each next term is (4) less than the previous one, so (d=-4). A decreasing arithmetic progression has a negative common difference.

Correct answer: B. 0

In an arithmetic progression, consecutive differences are equal: b-a=c-b. Rearranging gives a+c=2b. Hence, a-2b+c=a+c-2b=0. A common mistake is to focus only on the difference; the middle term is always the average of the first and third terms. Exam tip: for three AP terms, use 2b=a+c directly.

Correct answer: C. 3

In an arithmetic progression, the differences between consecutive terms are equal. Here, the first difference is \((2a+1)-(a+4)=a-3\), and the second difference is \((5a-8)-(2a+1)=3a-9\). Thus, \(a-3=3a-9\), which gives \(a=3\). Therefore, option 3 is correct. Exam tip: For three terms to be an AP, equate the first and second differences.

Correct answer: D. No such (x) exists

The differences are (x-1, 2x+3, 2x+3). Equating the first two gives (x=-4), but then all differences do not stay equal.

Correct answer: A. \(d=0\)

The consecutive differences are \((10+d)-10=d\), \((10+2d)-(10+d)=d\), and \((10+4d)-(10+2d)=2d\). In an arithmetic progression, all consecutive differences must be equal. Thus, \(d=2d\), which gives \(d=0\). If \(d=2\) or \(d=4\), the last difference becomes 4 or 8 respectively, so it is not equal to the earlier difference. Exam tip: To test an AP, compare the differences between consecutive terms.

Correct answer: B. \(2m\)

Find the differences between consecutive terms: \((3+2m)-3=2m\), \((3+4m)-(3+2m)=2m\), and \((3+6m)-(3+4m)=2m\). Since every consecutive difference is the same, the common difference is \(2m\). \(m\) is not the increase between consecutive terms; subtract consecutive terms to obtain the difference. Exam tip: In an AP, verify using \(d=a_{n+1}-a_n\).

Correct answer: A. For every (y)

In an arithmetic progression, the differences between consecutive terms must be equal. Here, second term − first term = (3y+10)−(5y+2)=−2y+8. Also, third term − second term = (y+18)−(3y+10)=−2y+8. Since the two differences are equal for every (y), the correct answer is every (y). Although (y=0) and (y=4) also work, the AP is not restricted to those values. Exam tip: For three terms in AP, twice the middle term equals the sum of the first and third terms.

Correct answer: C. 4

In an arithmetic progression, the differences between consecutive terms must be equal. Here, the first difference is \((2k+3)-(k+7)=k-4\), and the second difference is \((4k-5)-(2k+3)=2k-8\). Thus, \(k-4=2k-8\), which gives \(k=4\). Therefore, option C is correct. For instance, at \(k=3\), the two differences are not equal. Exam tip: For three terms to form an AP, equate the first and second consecutive differences.

Correct answer: D. No such common difference exists

In an arithmetic progression, the middle term is the average of the first and third terms. Therefore, \(2(2x+1)=13+(4x-5)\) must hold. Simplifying gives \(4x+2=4x+8\), or \(2=8\), which is impossible. Hence, for no value of \(x\) do these three terms form an AP, so no common difference exists. Exam tip: for three terms \(a,b,c\), check an AP quickly using \(2b=a+c\).

Correct answer: B. \(p=3,\ d=2\)

In an arithmetic progression, the differences between consecutive terms are equal. Here, the difference between the second and first terms is \((6p-3)-(4p+1)=2p-4\), while that between the third and second terms is \((9p-10)-(6p-3)=3p-7\). Thus, \(2p-4=3p-7\), giving \(p=3\). Substituting this value, \(d=2p-4=2\). For option C, putting \(p=4\) does not make the two differences equal. Exam tip: In AP questions, equate the two consecutive differences first.

Correct answer: B. (5, 9, 13, 20)

In (5,9,13,20), the first two differences are (4,4), but the third difference is (7). It is necessary to check all consecutive differences.

Correct answer: A. (0)

The consecutive differences are (r,2r,3r), which are equal only when (r=0). In such questions first write all consecutive differences.

Correct answer: B. (3)

In three consecutive terms, twice the middle term equals the sum of the other two terms. So (2(x+8)=(2x+1)+(3x-4)) gives (x=3).

Correct answer: B. (4)

(2(3a+6)=(5a-2)+(a+18)) gives (a=2). Then the terms (8,12,20) do not have equal differences, so no arithmetic progression is formed.

Correct answer: A. Only (p=0)

The direct answer is option A: only
\(p=0\). An arithmetic progression has the same difference between every pair of consecutive terms. Here the first differences are
\((4+p)-4=p\), then
\((4+3p)-(4+p)=2p\), and then
\((4+6p)-(4+3p)=3p\). For an AP, we need
\(p=2p=3p\). The first equality gives
\(p=0\), and this value also makes every difference zero, so the sequence becomes
\(4,4,4,4,\ldots\). Option A is correct. Option B,
\(p=1\), gives differences
\(1,2,3\), which are unequal. Option C is wrong because an arbitrary p does not make these differences equal. Option D is wrong because p=0 does work. The useful method is always to subtract consecutive terms before choosing an answer.

Correct answer: C. 13

Consecutive terms of an arithmetic progression have the same common difference. Here, the difference between 25 and 19 is 6, so the term immediately before 19 is 19 - 6 = 13. Therefore, u = 13. Option 12 is not correct because the difference from 12 to 19 would be 7, not the common difference 6. Exam tip: Find the difference between known consecutive terms and check it in both forward and backward directions.

Correct answer: B. (\frac{1}{4})

(\frac{5}{8}-\frac{3}{8}=\frac{2}{8}=\frac{1}{4}). When denominators are equal, subtract the numerators.

Correct answer: B. (-4)

Adding the same number to all terms does not change the common difference. The original (d=20-24=-4), so the new (d) remains (-4).

Correct answer: C. It becomes negative

The direct answer is option C: the common difference becomes negative. In the original sequence, moving forward gives
\(8-3=5\),
\(13-8=5\), and
\(18-13=5\), so the original common difference is
\(d=5\). Reversing the order gives
\(18,13,8,3,\ldots\). Now consecutive differences are
\(13-18=-5\),
\(8-13=-5\), and
\(3-8=-5\). Thus the new common difference is
\(-5\), which is negative. Option A is wrong because the direction has reversed, so +5 does not describe the new order. Option B is wrong because the terms are not equal. Option C is correct. Option D is wrong because a constant difference still exists. Memory cue: reversing an AP changes
\(d\) to
\(-d\).

Correct answer: C. True for every (x)

Both consecutive differences are (6) and (6), so it is an arithmetic progression for every (x). Simplify differences before deciding the variable.

Correct answer: A. \(4\)

The common difference of the original sequence is \(5-9=-4\). Taking the opposite of every term gives \((-9,-5,-1,3,\ldots)\). The difference between consecutive terms is \(-5-(-9)=4\), so the new common difference is \(4\). The value \(-4\) is the common difference of the original sequence, not the new one. Exam tip: when every term is multiplied by \(-1\), the sign of the common difference also changes.

Correct answer: A. (\frac{1}{5},\frac{4}{5},\frac{7}{5},2,\ldots)

In the first option, every consecutive difference is (\frac{3}{5}). In fraction options, check all differences separately.

Correct answer: D. No value

(20-2=18) is split into three gaps, so the second term should be (8) and the third (14). This gives (x=5) and (x=\frac{13}{3}), so no single value is possible.

Correct answer: C. 0.45

The common difference is found by subtracting a term from the next term. Here, \(1.05-0.60=0.45\) and \(1.50-1.05=0.45\), so the common difference is \(0.45\). Choosing \(0.40\) would not give the correct difference between consecutive terms. Exam tip: Align decimal points before subtracting decimal numbers.

Correct answer: B. 6

In an arithmetic progression, the common difference is the difference between consecutive terms. Here, \((r-3)-(r-9)=r-3-r+9=6\). Checking the next pair, \((r+3)-(r-3)=6\) as well. Therefore, \(d=6\). Option \(r\) is incorrect because the common difference must not depend on the variable. Exam tip: Find \(d\) by subtracting the first term from the second term.