Introduction to APs and common difference.

Check the differences between consecutive terms: \(6-3=3\), \(9-6=3\), and \(12-9=3\). Since the difference is always 3, the sequence is an arithmetic progression with common difference 3. Therefore, A is wrong because the terms are increasing, and D is wrong because 6 and 12 are even. Exam tip: A sequence is an AP when the differences between consecutive terms are equal.

The differences between consecutive terms are 5−2=3, 10−5=5, and 17−10=7. These differences are not equal, whereas an arithmetic progression must have the same difference between every pair of consecutive terms. Therefore, the sequence is not an arithmetic progression. Exam tip: Always check consecutive differences before identifying a sequence as an AP.

In an arithmetic progression, the next term is found by adding the common difference to the preceding term. Thus, second term = first term + common difference = 7 + 2 = 9. Option 8 would result from adding 1, not the given common difference 2. Exam tip: use the formula \(a_2=a_1+d\) for the second term.

The first term of the sequence is 20. The common difference is the second term minus the first term: 15 − 20 = −5, and this difference continues between consecutive terms. Therefore, the correct ordered pair is (20, −5). Option C has the wrong sign for the difference, while options B and D use an incorrect first term. Exam tip: “respectively” means that the answers must be given in the same order as the quantities asked.

The first term is a=4 and the common difference is d=6. Therefore, the terms are 4, 4+6=10, and 10+6=16, so the first three terms are (4, 10, 16). Option A is incorrect because it adds 4 instead of the common difference 6. Exam tip: Add the common difference to each term to obtain the next term.

The common difference is (3), so the next term is (10+3=13). The next term is found by adding (d) to the previous term.

The common difference is the difference between two consecutive terms. Here, \(d=0-(-3)=3\), and the next difference \(3-0=3\) confirms it. Therefore, the correct answer is 3. In an exam, subtract any two consecutive terms and handle the negative sign carefully.

The general term of an arithmetic progression is \(a_n=a_1+(n-1)d\). When the common difference is \(d=0\), we get \(a_n=a_1\), so every term is equal to the first term and all terms are equal. All terms will be zero only if the first term is also zero. Exam tip: Recognise an AP with \(d=0\) as a constant sequence.

The common difference is the difference between two consecutive terms. Since every consecutive term is 8, the common difference is \(8-8=0\). Therefore, option B is correct. Exam tip: A sequence with identical consecutive terms is constant, so its common difference is always 0.

In an arithmetic progression, the equal difference between every pair of consecutive terms is called the common difference and is usually denoted by (d). The first term is denoted by (a), while the number of terms is denoted by (n), so options A, B and D are incorrect. Exam tip: subtract each term from the next; the equal result is d.

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 satisfy two conditions: its terms must become smaller, and its common difference must be a constant negative number. In option A, the successive differences are 45 − 50 = −5, 40 − 45 = −5, and 35 − 40 = −5. Thus both conditions hold, so option A is correct. Option B has a constant ratio of 2 rather than a constant difference, making it geometric. Option C has common difference 0, so it is constant rather than decreasing. Option D consists of successive squares; its differences are 3, 5, and 7, which are not constant. Hence only option A is a decreasing arithmetic progression.

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.