Introduction to APs and common difference.

The differences between consecutive terms are equal: 4−2=2, 6−4=2, and 8−6=2. Therefore, the sequence is an arithmetic progression. It is not a geometric progression because a geometric progression must have a constant ratio between consecutive terms, which is not true here. Exam tip: To identify an arithmetic progression, check whether the differences between consecutive terms are equal.

In an arithmetic progression, each successive term is obtained by adding the common difference d to the preceding term. When d>0, each next term is larger than the previous one, so the terms increase. Option D would be correct only when d=0, while the terms decrease when d<0. Exam tip: check the sign of d—positive means increasing, zero means constant, and negative means decreasing.

In an arithmetic progression, each successive term is obtained by adding the common difference \(d\) to the previous term. When \(d<0\), a negative number is added each time, so the terms decrease successively. Option B would be correct only when \(d=0\). Exam tip: terms increase for \(d>0\), remain equal for \(d=0\), and decrease for \(d<0\).

The position of a term is found by counting from the beginning of the sequence, starting with position 1. In the sequence \(7,10,13,16,\ldots\), the number 7 is written first, so it is the first term. The next number, 10, is therefore the second term. This directly gives option B.

The common difference is 3 because each term increases by 3, but finding the difference is not necessary for this particular question. If we use the formula for an arithmetic progression, \(a_n=a+(n-1)d\), then \(a=7\) and \(d=3\). For \(n=2\), \(a_2=7+(2-1)3=10\). This confirms that 10 is the second term, not the first, third, or fourth.

The common difference of an arithmetic progression is the difference between two consecutive terms. Here, \(d=28-31=-3\), and the next difference is also \(25-28=-3\). Therefore, option B, \(-3\), is correct. In a decreasing arithmetic progression, the common difference is negative; check the sign by subtracting the preceding term from the next term.

In an arithmetic progression, the common difference is d = second term − first term. Here, d = 9 − 4 = 5, so option B is correct. The number 9 is the next term, not the common difference. In an exam, subtract the earlier term from the consecutive term to find d.

In an arithmetic progression, the difference between consecutive terms is constant. In option B, \(8-8=0\), and every consecutive difference is \(0\); therefore, its common difference is \(d=0\). Options A and C have common differences \(1\) and \(-2\), respectively, while the differences in option D are not constant. Exam tip: subtract each term from the next one to check whether the common difference is constant.

The governing definition is that a list or sequence is an arithmetic progression when subtraction of each term from the next gives the same constant value. For example, in 4, 7, 10, 13, the differences are 3, 3, and 3, so it is an arithmetic progression. A geometric progression is identified by a constant ratio, such as 2, 6, 18, where each term is multiplied by 3. A set is a collection of objects and does not require an ordered constant difference, while an equation is a statement that two expressions are equal. Therefore, option A gives the correct name.

In option A, the consecutive differences are \(5-2=3\), \(8-5=3\), and \(11-8=3\). Since the common difference is constant, it is an AP. In option D, the differences are \(-2,-3,-4\), so it is not an AP. Exam tip: compare consecutive differences.

The common difference is found by subtracting a term from the succeeding term. Here, \(10-12=-2\), and the same difference occurs between successive terms. Therefore, the common difference is \(-2\). Exam tip: a decreasing AP has a negative 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.