Introduction to APs and common difference.

In the standard notation for an arithmetic progression, \(a\) denotes the first term and \(d\) denotes the common difference. Therefore, the correct answer is the first term. Option A is incorrect because the common difference is usually represented by \(d\). In exams, remember the roles of \(a\) and \(d\) in the formula \(a_n=a+(n-1)d\).

This is an arithmetic progression because the difference between consecutive terms is constant. The common difference is found by subtracting the first term from the second: \(d=7-4=3\). Therefore, the correct answer is 3. In an exam, verify the result using any other pair of consecutive terms, such as \(10-7=3\).

In an arithmetic progression, the first term is generally denoted by \(a\). The first term of the given sequence is 11, so \(a=11\). The number 14 is the second term and 3 is the common difference, so neither is the value of \(a\). Exam tip: To find \(a\) in an AP, identify the first term of the sequence.

The first term is 10, and each subsequent term is obtained by adding d=-1 to the previous term. Thus, the terms are 10, 10-1=9, 9-1=8, and 8-1=7. Option B uses a common difference of +1, while option D uses -2. Exam tip: a negative common difference produces decreasing terms.

In option B, the differences between consecutive terms are equal: 9−2=7, 16−9=7, and 23−16=7. Therefore, its common difference is 7. Option A has a common difference of 5, while the consecutive differences in options C and D are not equal. Exam tip: subtract each term from the next one; in an AP, the difference must remain constant.

To find the common difference, subtract a term from the term immediately before it: \(90-100=-10\). Therefore, the correct answer is \(-10\). Option A gives only the magnitude of the difference, but a decreasing sequence has a negative common difference. Exam tip: calculate second term − first term using any two consecutive terms.

In an arithmetic progression, the difference between consecutive terms is constant. Therefore, each next term is obtained by adding the same fixed number to the previous term; this number is called the common difference. For example, in 3, 7, 11, 15, the common difference is 4. Multiplication or division between consecutive terms may describe a geometric progression, while the sum of the terms is not always 0. Exam tip: subtract consecutive terms; if all the differences are equal, the sequence is an AP.

The first term of the sequence is \(a=15\). The difference between consecutive terms is constant: \(18-15=3\), so the common difference is \(d=3\). Therefore, \((a,d)=(15,3)\), making option A correct. Exam tip: In an AP, \(a\) denotes the first term and \(d\) is the difference between the second and first terms.

This is an arithmetic progression with common difference 5: \(5-0=10-5=15-10=5\). Therefore, the next term is \(15+5=20\). Option 25 is incorrect because it would require adding 10 to 15. Exam tip: To find the next term of an AP, add the common difference to the last given term.

An arithmetic progression is a sequence in which the difference between every pair of consecutive terms is the same constant. The terms may increase or decrease, but equal spacing is the essential requirement. In the sequence \(1,3,6,10,\ldots\), the first difference is \(3-1=2\), the next is \(6-3=3\), and the next is \(10-6=4\). Since these differences are not equal, the sequence is not an arithmetic progression.

Therefore, option C is correct. The fact that the terms increase does not by itself make a sequence an AP; for example, increasing differences can still occur. The first term being 1 is irrelevant, and an AP does not need a displayed final term because it may continue indefinitely. The decisive test is equality of consecutive differences.

In an arithmetic progression, the common difference is the difference between two consecutive terms. Thus, \(d=5-2=3\), so option D is correct. Option A is the first term, while option B is the next term. Exam tip: use \(d=a_2-a_1\) for two consecutive terms.

A decreasing arithmetic progression must have a constant negative common difference. In option A, the consecutive differences are 45 − 50 = −5, 40 − 45 = −5, and 35 − 40 = −5. Both conditions are satisfied: the terms decrease and the difference remains constant, so option A is correct. Option B doubles each time, giving a constant ratio rather than a constant difference, so it is geometric. Option C has common difference 0, so it is constant, not decreasing. Option D contains successive squares, whose differences are 3, 5, and 7, so the difference is not constant.

The common difference is found by subtracting a term from the term immediately after it. Here, d = 3.0 - 2.5 = 0.5 and 3.5 - 3.0 = 0.5, so the common difference is 0.5. The value 2.5 is the first term, not the difference. Exam tip: In an AP, subtract any term from the next consecutive term.

In an AP, the difference between every pair of consecutive terms must be constant. Here the differences are \(4,4,5\), so it is not an AP. Exam tip: check differences, not merely whether terms increase.

In an arithmetic progression, the first term is a and the common difference d is added to obtain each successive term. Thus, in the sequence a, a+d, a+2d, ..., the second term is a+d. The expression a+2d represents the third term, not the second. Exam tip: remember the first few terms as a, a+d, a+2d, ... to identify the required term quickly.

The terms of an arithmetic progression are obtained by repeatedly adding the common difference d to the first term a: the first term is a, the second is a+d, and the third is a+2d. Therefore, the correct answer is a+2d. Remember that d is added twice for the third term; it is not multiplied by a.

The difference between consecutive terms is constant: 4.5−4=0.5, 5−4.5=0.5, and 5.5−5=0.5. Thus, the common difference is 0.5, so the next term is 5.5+0.5=6. Option A is incorrect because it adds only 0.25. In an exam, first find the common difference and then add it to the last given term.

An arithmetic progression is a sequence in which the difference between consecutive terms remains constant. In the sequence \(3,3,3,3,\ldots\), subtracting one term from the next gives \(3-3=0\) every time. Therefore the common difference is \(d=0\). The fact that the terms do not increase or decrease does not disqualify the sequence from being an arithmetic progression.

The first term is \(a=3\), not zero. Thus the correct statement is option B: it is an arithmetic progression with common difference zero. A constant sequence is an important special case of an arithmetic progression. The value 3 is the repeated term, while 0 describes the change between successive terms.

The common difference of an arithmetic progression is the difference between two consecutive terms. Thus, \(d=11-14=-3\). This is verified by \(8-11=-3\) and \(5-8=-3\). Therefore, -3 is correct. Option A results from missing the negative sign. Exam tip: calculate the common difference as the second term minus the first term.

There is an equal increase of (6) each time, so it is an arithmetic progression. In word problems look for equal increase.