Introduction to APs and common difference.

Correct answer: D. (p+s=q+r)

Writing the terms as (p,p+d,p+2d,p+3d) gives (p+s=q+r). In exams, verify four-term relations symbolically.

Correct answer: D. (b-a=c-b)

In an AP, consecutive differences are equal, so (b-a=c-b). In exams, the same condition can be written as (2b=a+c).

Correct answer: A. (11, 9, 7, 5)

In an AP, the common difference d is added to each term to get the next term. Here d = -2, so starting from 11 gives 9, 7, and 5. Option B has a common difference of +2, so it is incorrect. Exam tip: a negative d means that the terms decrease successively.

Correct answer: C. (-3)

Multiplying by (\frac{1}{3}) changes the difference from (-9) to (-3), and adding (5) does not change it. In exams, separate the effects of addition and multiplication.

Correct answer: C. (12)

The new sequence jumps three original terms each time, so its difference is (3d=12). In exams, count the gap between selected term numbers.

Correct answer: D. (6)

The new common difference is 6, so D is correct. In the original finite AP, moving forward one place changes a term by d=−6. When the terms are written backward, moving forward in the reversed list means moving backward in the original list; therefore each change is the opposite, −d=−(−6)=6. A, −12, doubles the magnitude incorrectly. B, −6, is the old difference and does not account for reversal. C, 0, would mean all terms are equal, which is not true when the original difference is −6. D, 6, is the correct sign reversal. For example, 20,14,8 becomes 8,14,20, whose differences are 6. Memory cue: reversing a finite AP changes d to −d, not its magnitude to a new value.

Correct answer: A. (1)

The condition (2r^2=r+r^3) gives (r(r-1)^2=0), so the nonzero value is (1). In exams, always apply the nonzero condition.

Correct answer: C. (7)

For three AP terms, (2\beta=\alpha+\gamma), so (\beta=7). In exams, treat the middle term as the average of the extremes.

Correct answer: D. (m+1)

(a_n=(m+1)n), so the coefficient of (n) is (m+1). In exams, combine like terms before identifying the linear coefficient.

Correct answer: C. It is not an AP because the differences are (2,3,4,5,\ldots)

The consecutive differences keep increasing, so they are not constant. In exams, a famous pattern need not be an AP.

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

(\frac{5}{8}-\frac{7}{8}=-\frac{2}{8}=-\frac{1}{4}). In exams, with equal denominators, subtract numerators directly.

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

The middle-term condition gives (x^2-4=x^2), which is impossible. In exams, also check that denominators are nonzero.

Correct answer: C. No, the differences are (4,12,36,\ldots)

The consecutive differences are not equal, so it is not an AP. In exams, do not confuse a multiplicative pattern with an AP.

Correct answer: A. (2\lambda+6)

The common difference is second term minus first term, so (3\lambda+2-(\lambda-4)=2\lambda+6). In exams, handle the minus sign before brackets carefully.

Correct answer: B. No, the consecutive difference changes

The initial terms are (1,5,5,9,\ldots), so the difference is not constant. In exams, test terms containing ((-1)^n) by substituting a few values.

Correct answer: B. (12) kilometres

The distance increases by (12) kilometres each time. In word problems, still find the same consecutive difference.

Correct answer: B. (k=2)

The differences are (k+1,3k-3,3k-3), and equality gives (k=2). In exams, check all consecutive differences for four terms.

Correct answer: A. (0)

Here (c_n=4) for every (n), so its common difference is (0). In exams, treat the sequence of differences as a separate sequence.

Correct answer: C. Only (a_2-a_1=a_3-a_2)

Checking only the first three terms is not enough for the whole sequence. In exams, the full condition must involve every (n) or all consecutive terms.

Correct answer: A. 7 दिन

Find the difference between consecutive terms: \(10-3=7\), \(17-10=7\), \(24-17=7\), and \(31-24=7\). Therefore, this is an AP with common difference \(7\) days. \(14\) days represents two weeks, but each successive date here is only 7 days later. Exam tip: To find the common difference, subtract the first term from the second term.

Correct answer: D. No, the consecutive difference is not constant

The terms are (\frac{1}{3},\frac{1}{5},\frac{1}{7},\ldots), and the differences change. In exams, test fractions with changing denominators by differences.

Correct answer: B. (2d)

The new sequence jumps two original terms each time, so the difference is (2d). In exams, multiply (d) by the term-number jump.

Correct answer: B. (9)

Moving two terms ahead changes the value by (2d), so (2d=18) and (d=9). In exams, count the number of jumps.

Correct answer: C. (7)

There are (5) intervals between (a_7) and (a_2), so (5d=35) and (d=7). In exams, subtract term numbers to get intervals.

Correct answer: B. (-2)

The sum sequence has difference (3+(-5)=-2). In exams, for sums of corresponding terms, add the differences too.