Introduction to APs and common difference.

Correct answer: C. \(0, -4\)

In an AP, \(a\) is the first term, so \(a=0\). The common difference is \(d=a_2-a_1=-4-0=-4\). Hence, the correct pair is \(0,-4\). In option B, \(-4\) is the second term, not the first term. Exam tip: To find \(d\), subtract a term from the term immediately following it.

Correct answer: C. 0.8

The first term is 1.2 km and the third term is 2.0 km. Therefore, the difference is 2.0 - 1.2 = 0.8 km. There are two equal gaps from the first term to the third term, so the difference is 2d = 2 × 0.4 = 0.8. Option 0.4 is only the difference between consecutive terms. Exam tip: In an AP, use aₙ - aₘ = (n - m)d.

Correct answer: B. (6)

Adding the same (3) to each term does not change the difference, so (d=6). Equal change in all terms does not change (d).

Correct answer: A. (-2)

The common difference is (-7-(-12)=5), so the third term is (-7+5=-2). Watch signs carefully with negative numbers.

Correct answer: C. -4

The common difference is found by subtracting a term from the term immediately after it. Here, \(d=(3a-4)-3a=-4\). Also, \((3a-8)-(3a-4)=-4\), confirming that the common difference is \(-4\). The number \(4\) is only the magnitude of the difference; for a decreasing AP, the common difference is negative. Exam tip: Write \(d=a_2-a_1\) and check the sign carefully.

Correct answer: C. 22

In three consecutive terms of an arithmetic progression, the middle term is the arithmetic mean of the first and third terms. Therefore b = (13 + 31)/2 = 44/2 = 22. Equivalently, equal successive differences require b − 13 = 31 − b; solving gives 2b = 44 and b = 22. Hence option C is correct. With b = 22, both differences are 9: 22 − 13 = 9 and 31 − 22 = 9. The other options do not make the two consecutive differences equal; for example, b = 18 gives differences 5 and 13. The middle-term average rule applies because the three values are consecutive AP terms.

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

In (\frac{1}{4},1,\frac{7}{4},\frac{5}{2},\ldots), every difference is (\frac{3}{4}). Check fractions using common denominators.

Correct answer: C. 12

The second term of an AP is \(a_2=a+d\). Here, \(17=5+d\), so \(d=17-5=12\). Therefore, 12 is correct. Note that 17 is the second term, not the common difference. Exam tip: find \(d\) by subtracting a term from the following term.

Correct answer: C. (0.9)

The common difference of an arithmetic progression is the fixed amount added to one term to obtain the next term. It is found by subtracting any term from the term immediately after it. In this sequence, the numbers rise regularly, so the same difference should appear between every pair of consecutive terms. This is the central idea behind identifying an AP and its common difference.

Subtract the first term from the second: \(3.1-2.2=0.9\). Checking the next pair gives \(4.0-3.1=0.9\), and the following pair gives \(4.9-4.0=0.9\). Thus the fixed common difference is \(d=0.9\), so option C follows. Options 0.7, 0.8, and 1.1 do not match the consecutive gaps.

Correct answer: C. (21-28=-7)

The common difference is next term minus previous term, so (21-28=-7). In a decreasing sequence the order of subtraction is very important.

Correct answer: B. \(11\)

In an AP, the second term is \(a_2=a_1+d\). Therefore, \(a_1=a_2-d=20-9=11\). Hence, the first term is \(11\). The value \(20\) is the second term, not the first term. Exam tip: To find the first term from the second term, subtract the common difference.

Correct answer: C. (10)

The original common difference is (5), and after multiplying by (2), the new common difference becomes (10). When all terms are multiplied by the same number, (d) is also multiplied by it.

Correct answer: B. (-6)

The reverse order is (24,18,12,6,\ldots), and (d=18-24=-6). When the order is reversed, the sign of the common difference changes.

Correct answer: B. (x=10,y=15)

There are three equal gaps from (5) to (20), so (d=\frac{20-5}{3}=5), hence (x=10) and (y=15). Find missing terms by splitting the total difference into equal gaps.

Correct answer: A. (2, 6, 10, 15, …)

The defining condition for an arithmetic progression is that every consecutive difference must be the same, not merely the first one or two. In option A, the first differences are 6 − 2 = 4 and 10 − 6 = 4, but the next difference is 15 − 10 = 5. Because the difference changes from 4 to 5, this sequence is not an AP. In option B, all displayed differences are 5; in option C, they are −4; and in option D, they are 0. Therefore option A is the only sequence that fails the definition. This question emphasizes why all available consecutive differences should be checked before classifying a sequence.

Correct answer: C. 10

The fourth term is 3+3d, so the given condition gives 3+3d=24. Hence 3d=21 and d=7. The second term is 3+d=3+7=10. Therefore option C is correct. The other choices do not satisfy the same value of d: using 8, 9 or 12 as the second term would make the fourth term different from 24.

Correct answer: D. 4

For three numbers to be consecutive terms of an arithmetic progression, the middle term is the arithmetic mean of the first and third. Thus twice the middle term equals the sum of the outer terms: 2(3u + 2) = (2u − 1) + (5u + 1). Expanding gives 6u + 4 = 7u, so u = 4. Substitution provides a complete check: the three terms become 7, 14, and 21. Their consecutive differences are 14 − 7 = 7 and 21 − 14 = 7, so they do form an AP. Therefore option D is correct. The other listed values do not make the two consecutive differences equal and hence cannot satisfy the AP condition.

Correct answer: C. 6

The governing concept is the middle-term condition for three consecutive terms of an arithmetic progression: twice the middle term equals the sum of the first and third terms. Therefore, 2(k + 2) = (k − 3) + (2k + 1). On simplifying, 2k + 4 = 3k − 2, so k = 6. Substitution provides an independent check: the three terms become 3, 8, and 13. Their consecutive differences are 8 − 3 = 5 and 13 − 8 = 5, so the difference is constant and the sequence is an AP. Hence option C is correct. Values such as 5 or 7 can result from an algebraic sign or rearrangement error, but they do not make both consecutive differences equal.

Correct answer: D. 3

The statement becomes false precisely when the three displayed terms do form an arithmetic progression. For three terms, apply the condition that twice the middle term equals the sum of the first and third terms: 2(3m − 2) = (m + 1) + (6m − 8). This gives 6m − 4 = 7m − 7, and hence m = 3. Substitution confirms the result: the terms are 4, 7, and 10, whose consecutive differences are both 3. Thus the original claim that the sequence is not an AP is false, and option D is correct. The values 1, 2, and 4 do not produce equal consecutive differences. The phrase “become false” is important: we are seeking the value that makes the AP condition true.

Correct answer: C. 9

In a four-term arithmetic progression, the change from the first term to the fourth term covers three equal common differences. Let that common difference be d. The fourth term is therefore 4 + 3d, and the given fourth term gives 4 + 3d = 31. Hence 3d = 27 and d = 9. Since r is the second term and s is the third term, the difference s − r is exactly one common difference, so s − r = 9. A full check gives the sequence 4, 13, 22, 31, where every adjacent difference is 9. Therefore option C is correct. The value 27 is the total change from the first to the fourth term, not the difference between adjacent terms; confusing these quantities is the main likely error.

Correct answer: B. Common difference

Calculate the two consecutive terms using the given rule. A_15=4(15)−9=51 and A_14=4(14)−9=47, so A_15−A_14=51−47=4. More generally, A_(n+1)−A_n=[4(n+1)−9]−[4n−9]=4. Since this value is constant, the sequence is an arithmetic progression, and the difference between any two consecutive terms is its common difference. Therefore option B is correct. The expression does not represent the first term, because the first term would be A_1; it does not represent the number of terms, because no finite list is specified; and it does not represent a last term, because A_15−A_14 is a difference, not a term. The governing idea is that a linear nth-term formula A_n=an+b has common difference a.

Correct answer: A. 19

For three consecutive terms of an arithmetic progression, twice the middle term equals the sum of the first and third terms. Apply this condition: 2(x+6)=x+(3x−2). After expansion, 2x+12=4x−2, so 14=2x and x=7. The third term is therefore 3x−2=3(7)−2=21−2=19. Hence option A is correct. A direct check confirms the result: the terms become 7, 13, 19, whose consecutive differences are 6 and 6. Option C is the unadjusted value 3x after finding x=7, while the other numerical choices do not satisfy the defining AP condition. The governing concept is equality of consecutive differences, equivalently the middle-term relation 2b=a+c.

Correct answer: A. 8 and 8

Find each requested difference by subtracting consecutive terms. For the first two terms, (x+3)−(x−5)=x+3−x+5=8. For the last two terms, (x+19)−(x+11)=x+19−x−11=8. In both calculations, the variable x cancels, leaving the same constant difference. Thus the sequence has common difference 8, and the answer is option A. This also verifies the arithmetic-progression property: every successive difference is 8, including (x+11)−(x+3). Options C and D incorrectly retain x after cancellation, while option B gives two numbers that do not result from either subtraction. The key concept is that algebraic expressions may represent fixed differences even when each individual term contains a variable.

Correct answer: D. It is not an arithmetic progression for any x

For three terms to be in an arithmetic progression, twice the middle term must equal the sum of the first and third terms: 2(2x+1)=13+(4x-5). Simplifying gives 4x+2=4x+8, hence 2=8, which is impossible for every x. Therefore no value of x makes the sequence an AP, so option D is correct.

Correct answer: D. 5

Use the defining relation for three consecutive AP terms: twice the middle term equals the sum of the first and third terms. Thus 2(a+4)=(2a−5)+(4a−7). Simplifying gives 2a+8=6a−12, so 20=4a and a=5. Substituting a=5 produces the terms 5, 9, and 13. Their consecutive differences are 9−5=4 and 13−9=4, confirming that the terms really form an arithmetic progression. Therefore option D is correct. The other values do not make the two consecutive differences equal. The governing concept is the middle-term property of an AP, which converts the progression condition into a linear equation and gives a unique value of the variable.