Introduction to APs and common difference.

Correct answer: A. It is an AP and (d=r)

Subtracting each term from the next gives (r). In exams, apply the same difference rule even with symbols.

Correct answer: D. It is an AP and (d=6)

The consecutive difference is the constant (6), so the sequence is an AP. In exams, a constant (a_{n+1}-a_n) identifies an AP.

Correct answer: C. No, (a_{n+1}-a_n=2n-2) is not constant

Here the difference depends on (n), so it is not constant. In exams, a quadratic (a_n) usually does not form an AP.

Correct answer: B. 4

In an AP, the difference between two terms equals the difference in their positions multiplied by the common difference. Thus, \(a_9-a_5=(9-5)d\), so \(34-18=4d\). Hence \(16=4d\) and \(d=4\). Option 16 is the difference between the two given terms, not the common difference. Exam tip: find the difference in term numbers first, then divide the term difference by it.

Correct answer: C. Every real (k)

Both differences are (2k-3), so it forms an AP for every real (k). In exams, simplify both differences first.

Correct answer: D. (\frac{1}{2},\frac{3}{4},1,\frac{5}{4})

An arithmetic progression must have the same difference between every two consecutive terms. The required difference here is \(\frac14\), so subtract each term from the next and check whether the result is always \(\frac14\). Option D follows the pattern of adding one-fourth at every step, so it is the correct sequence.

For option D, \(\frac34-\frac12=\frac14\), \(1-\frac34=\frac14\), and \(\frac54-1=\frac14\). All consecutive differences are equal. In option A, the differences are not all equal; option B also has changing differences, and option C doubles irregularly. Therefore only option D is an AP with common difference \(\frac14\).

Correct answer: B. (-30^\circ)

Each next angle decreases by (30^\circ), so the difference is (-30^\circ). In exams, keep the unit in the answer.

Correct answer: C. (-\frac{2}{5})

The coefficient of (n) is (-\frac{2}{5}), which is the common difference. In exams, identify the form (a_n=\alpha n+\beta).

Correct answer: A. (a_1=11,d=-3)

Putting (n=1) gives the first term (11), and each step adds (-3). In exams, recognize the form (a+d(n-1)).

Correct answer: A. It is an AP, (d=\sqrt{2})

The terms become (\sqrt{2},2\sqrt{2},3\sqrt{2},4\sqrt{2}), so the difference is (\sqrt{2}). In exams, simplify radicals first.

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

Equal differences give (5-m=3m-7), so (m=3) and (d=2). In exams, be careful with signs in terms containing variables.

Correct answer: B. (2b=a+c)

In an AP, the middle term is the average of the two extremes, so (2b=a+c). In exams, this is the fastest check for three terms.

Correct answer: D. (1, 4, 7, 11)

In an arithmetic progression, the difference between every pair of consecutive terms must be the same. In option D, the consecutive differences are 3, 3, and 4, which are not equal; hence it is not an AP. In options A, B, and C, the common differences are 3, 4, and 0 respectively, all constant. Exam tip: check every consecutive difference, not only the first two.

Correct answer: B. (15)

The amount increases by (15) rupees each time, so (d=15). In word problems, still look for the consecutive difference.

Correct answer: B. (k=1)

The condition (2k^2=k+k^3) gives (k(k-1)^2=0), and the nonzero value is (1). In exams, do not ignore conditions like nonzero.

Correct answer: D. No, there is no valid (p)

The middle-term condition leads to an impossible equation, so there is no valid (p). In exams, also check that denominators are nonzero.

Correct answer: D. Every real (h)

Both differences are (h), so it is an AP for every real (h). In exams, remember that (h=0) gives a valid constant AP.

Correct answer: B. (a_n=3n+1)

(3n+1) is linear and has difference (3). In exams, a linear form with constant coefficient gives an AP.

Correct answer: D. 7

In an AP, \(a_3=a_1+2d\). Therefore, \(a_3-a_1=2d=14\), so \(d=7\). The value \(-7\) would apply only if the given difference were \(-14\). Exam tip: use \(a_m-a_n=(m-n)d\) and carefully count the difference between the term numbers.

Correct answer: D. (d) cannot be determined

In an AP, (a_2+a_4=2a_3) is automatically true, so (d) is not found. In exams, check whether the information is new or just an identity.

Correct answer: A. (a_{n+1}-a_n) is constant for every (n)

The definition of an AP is based on a constant consecutive difference. In definition-based questions, use this test.

Correct answer: C. It is an AP and (d=0)

Every consecutive difference is (0), so it is a constant AP. In exams, treat (d=0) as a valid common difference.

Correct answer: C. No, there is no valid (x)

The middle-term condition gives (x(x+2)=(x+1)^2), which is impossible. In exams, check validity after forming the equation.

Correct answer: D. (\alpha)

(a_{n+1}-a_n=\alpha), so the common difference is (\alpha). In exams, this is the key shortcut for a linear sequence.

Correct answer: A. (a+e=b+c)

Taking the terms as (a,a+d,a+2d,a+3d), both sides become (2a+3d). In exams, test relations by writing symbolic AP terms.