Introduction to APs and common difference.

Correct answer: B. (2)

In three consecutive terms, twice the middle term equals the sum of the other two, so (2(2x+5)=(3x-2)+(x+16)) gives (x=2). The middle-term rule is a fast exam method.

Correct answer: C. (5)

Using (2(3p+7)=(4p+1)+(p+19)) gives (p=6), so the terms become (25,25,25) and (d=0). First find the variable and then check (d).

Correct answer: D. (34)

The total difference (29-5=24) is split into three equal gaps, so (d=8), (x=13), and (y=21). For two missing terms, divide the total difference into equal gaps.

Correct answer: B. (12)

The original (d=4); multiplying by (3) makes (d=12), and subtracting the same (2) does not change (d). Equal addition or subtraction does not change (d).

Correct answer: C. (-1)

The original (d=-3), and the difference of (2n) is (2), so the new (d=-3+2=-1). For term-number changes, add or subtract its difference.

Correct answer: A. Only (r=0)

The consecutive differences are (r,3r,5r), which are equal only when (r=0). In such questions it is necessary to write all consecutive differences.

Correct answer: A. (4)

The original (d=-4), and in reverse order (6,10,14,18,\ldots), (d=4). Reversing the order changes the sign of (d).

Correct answer: C. (8)

The first sequence has (d=5) and the second has (d=3), so the sum sequence has (d=8). In termwise addition, common differences add.

Correct answer: A. (-11)

The first sequence has (d=-6) and the second has (d=5), so the difference sequence has (d=-6-5=-11). In termwise subtraction, the common differences also subtract.

Correct answer: B. \(\frac{3}{4}\)

\(-1-\left(-\frac{7}{4}\right)=\frac{3}{4}\), and the next difference is also \(\frac{3}{4}\). Be careful while subtracting negative fractions.

Correct answer: C. 0.55

In an arithmetic progression, the common difference d is the difference between consecutive terms. Here, d = 1.35 - 0.80 = 0.55. Checking further, 1.90 - 1.35 = 0.55 and 2.45 - 1.90 = 0.55. Therefore, the correct answer is 0.55; 0.50 is not the actual difference between the decimal terms. Exam tip: Align decimal points while subtracting decimals.

Correct answer: B. \(6\)

In an arithmetic progression, the common difference is the difference between consecutive terms. Here, \((u-2)-(u-8)=6\) and \((u+4)-(u-2)=6\). Thus, 6 is added to each successive term, so \(d=6\). \(u+6\) may be a next term, not the common difference. Exam tip: quickly check \(d=a_2-a_1\).

Correct answer: C. (7)

From the third to the seventh term there are (4) gaps, so (4d=28) and (d=7). Convert term distance into number of gaps.

Correct answer: B. (-6)

From the second to the fifth term there are (3) gaps, so (3d=-18) and (d=-6). In a decreasing progression, (d) remains negative.

Correct answer: C. (d+2)

The new terms are (a+2, a+d+4, a+2d+6), and both differences are (d+2). The difference (2) of the added numbers is also added to (d).

Correct answer: B. (5)

The original (d=6), and (n) has difference (1), so the new (d=6-1=5). When subtracting term number, its difference is also subtracted.

Correct answer: A. (4, 9, 14, 20,\ldots)

An arithmetic progression must have the same difference between every consecutive pair of terms. Matching only the first two differences does not prove that a sequence is an AP; later terms must also be checked. A single changed difference is enough to disqualify the sequence.

In choice A, the first difference is \\(9-4=5\\), and the second is \\(14-9=5\\). However, the third difference is \\(20-14=6\\), not 5. Therefore the difference is not constant, so choice A is not an arithmetic progression. In B the difference is always 5, in C it is always \\(-6\\), and in D it is always 0. Thus A is the only correct choice.

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 fixing the variable.

Correct answer: D. No value

(2(x+9)=2x+18) gives an identity, so it is an arithmetic progression for every (x). Therefore no single value of (x) is obtained.

Correct answer: B. (-2)

The original (d=6), and multiplying by (-\frac{1}{3}) gives new (d=-2). The multiplier applies directly to the common difference too.

Correct answer: C. Not an arithmetic progression because differences are not equal

The new terms are (9,49,121), and the differences are (40,72), which are not equal. Squaring generally does not preserve an arithmetic progression.

Correct answer: A. (15,\frac{25}{2},10,\frac{15}{2},\ldots)

(15-\frac{5}{2}=\frac{25}{2}), and (-\frac{5}{2}) continues to be added. Check both (a) and (d) conditions together.

Correct answer: C. (4)

From (2(4v+1)=(3v-4)+(6v-1)), (8v+2=9v-5), so (v=7). Use the twice-middle-term rule to find the variable.

Correct answer: A. (-1)

The original (d=3), and (4n) has (d=4), so the new (d=3-4=-1). In term-number based changes, change the differences too.

Correct answer: D. (70,62,54,46,\ldots)

The first three options have (d=-9), but (70,62,54,46,\ldots) has (d=-8). Check both value and sign in the options.