Introduction to APs and common difference.

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.

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.

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

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

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.

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.

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

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.

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.

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

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.

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.

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.

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.

Twice the middle term equals the sum of the other two terms, so (2(x+4)=(2x-3)+(3x-1)) gives (x=3). For three consecutive terms, the middle-term rule is fast.

From (2(2a+1)=(a-2)+(5a-4)), (4a+2=6a-6), so (a=4), and (d=9-2=7). In such questions find the variable first and then find (d).

The consecutive differences are (p,2p,3p), which are equal only when (p=0). Comparing differences is the safest method.

The difference from (16) to (22) is (6), so (y=16-6=10). Use the same difference both forward and backward for missing terms.

There are three equal gaps, so (d=\frac{24-3}{3}=7), hence (x=10) and (y=17). For two missing terms, split the total difference into equal gaps.

(\frac{7}{10}-\frac{2}{5}=\frac{3}{10}) and (1-\frac{7}{10}=\frac{3}{10}). Do not forget to use common denominators in fractions.