Introduction to APs and common difference.

The middle term is (y=\frac{-11+5}{2}=-3), and (d=8). In exams, be careful while averaging negative numbers.

Here the common difference is (-3), so the terms are decreasing. In exams, a decreasing AP has negative (d).

Equating differences gives (-2q-6=-2q+14), which is impossible. In exams, cancellation of the variable can produce a contradiction.

The signs alternate and the difference does not remain the same. In exams, test alternating-sign sequences by differences only.

Multiplying terms by (2) multiplies the difference by (2), so it becomes (8). In exams, adding or subtracting a constant does not change the difference, but multiplication does.

Adding the same number to every term does not change consecutive differences. In exams, a uniform translation leaves the difference unchanged.

Multiplication scales all differences by the same factor, so the new difference is (-3d). In exams, apply the transformation to the difference.

Subtracting the same number from every term does not change the difference. In exams, treat uniform subtraction as changing only the first term.

The consecutive differences are (v) and (v), so they are in AP. In exams, recognize the balanced form (u-v,u,u+v) quickly.

Both differences are (3x-4), so the terms form an AP for every real (x). In exams, if both differences are identical expressions, no separate solving is needed.

(\frac{1}{6}-\frac{2}{3}=-\frac{1}{2}), and the same difference continues. In exams, use common denominators with negative fractions.

The first terms are (2,\frac{3}{2},\frac{4}{3},\ldots), and the differences change. In exams, test fractional forms in (n) by differences.

In the reversed order, each step subtracts (d), so the new difference is (-d). In exams, reversing the order can change the sign of the difference.

The coefficient of (n) is (-5), so it is the common difference. In exams, first check the coefficient of (n) in a linear term.

The consecutive differences are not equal, so it is not an AP. In exams, test constancy of differences, not just the visible pattern.

Equal differences give (-3t+9=-2t+11), so (t=-2) and (d=15). In exams, subtract first from second and second from third.

The middle term is the average of the extremes, so (z=\frac{-17+23}{2}=3) and (d=20). In exams, handle signs carefully when averaging negatives.

The difference depends on (n), so it is not constant for all (n). In exams, if (n) remains in the difference, it is not an AP.

The coefficient of (n) is (\frac{5}{4}). In exams, the same rule works for linear formulas with fractions.

The direct answer is option C,
\(-3\). In an AP, the change between two terms equals the number of equal steps multiplied by the common difference. From the fourth term to the tenth term there are
\(10-4=6\) steps. The total change is
\(a_{10}-a_4=-6-12=-18\). Therefore
\(6d=-18\), so dividing by 6 gives
\(d=-3\). Option A,
\(-6\), forgets that there are six steps. Option B,
\(-4\), does not satisfy
\(6d=-18\). Option C is correct. Option D,
\(3\), has the wrong sign: a term falls from 12 to -6, so the common difference must be negative. Memory cue: subtract term values, count the gap in indices, then divide.

Equating differences gives (x+5=3x-8), so (x=\frac{13}{2}) and (d=\frac{23}{2}). In exams, do not reject a fractional answer too quickly.

In that option, each step increases by (0.75). In exams, use consecutive difference as the test even with decimals.

Each step decreases by (3.5^\circ), so (d=-3.5^\circ). In exams, use a negative sign for a decreasing sequence.

The direct answer is D: \(a_1=c\) and \(d=-r\). Compare the formula with the standard arithmetic-progression form \(a_n=a_1+(n-1)d\). Here \(a_n=c-(n-1)r=c+(n-1)(-r)\). Therefore the first term is c and the common difference is -r. Direct checking gives \(a_1=c-(1-1)r=c\). Also, subtracting consecutive terms gives \(a_{n+1}-a_n=-r\), confirming the difference. Option A incorrectly says the first term is c-r and gives a positive difference. Option B has the correct first term but wrong sign for the difference. Option C swaps the roles of c and r. Option D matches both results. Memory cue: the sign before r is the sign of the common difference.

The terms become (\sqrt{3},2\sqrt{3},3\sqrt{3},4\sqrt{3}). In exams, simplify radicals before finding differences.