Introduction to APs and common difference.

Correct answer: A. All terms will be equal

In an arithmetic progression, the difference between consecutive terms is constant and is denoted by \(d\). When \(d=0\), no change occurs from one term to the next, so the progression is \(a, a, a, a, \ldots\). Hence, all terms are equal. The terms need not all be negative; that depends on the first term. Exam tip: Recognise an AP with \(d=0\) as a constant progression.

Correct answer: B. (3, 8, 13, 18, \ldots)

In option B, the differences between consecutive terms are equal: \(8-3=5\), \(13-8=5\), and \(18-13=5\). Thus, its common difference is 5, so it is an arithmetic progression. In option A and option D, the terms are multiplied by 2, while option C has differences 3, 5, and 7; therefore, none of them is an arithmetic progression. For exams, calculate the differences between consecutive terms to identify an AP quickly.

Correct answer: B. 4

The common difference of an arithmetic progression is found by subtracting a term from the term immediately before it. Here,
(-6)-(-10)=-6+10=4. The next differences are also
(-2)-(-6)=4 and 2-(-2)=4, so the common difference is 4. Exam tip: when subtracting a negative number, its sign changes to plus.

Correct answer: A. 5

The common difference of an arithmetic progression is the difference between two consecutive terms. Here, the next term is 23 and the previous term is 18, so the common difference is 23 − 18 = 5. Therefore, option A is correct. Exam tip: use common difference = next term − previous term.

Correct answer: C. No, because the differences between consecutive terms are not equal

The differences between consecutive terms are 8−4=4, 13−8=5, and 19−13=6. In an arithmetic progression, the difference between every pair of consecutive terms must be constant. Since these differences are 4, 5, and 6, the sequence is not an arithmetic progression. Merely having increasing or positive terms is not sufficient. Exam tip: calculate consecutive differences to test whether a sequence is an AP.

Correct answer: C. 3

To find the common difference, subtract each term from the term immediately after it. Here, \(4-1=3\), \(7-4=3\), and \(10-7=3\). Therefore, each term is 3 greater than the previous term, so the common difference is 3. Exam tip: In an AP, the difference between consecutive terms remains constant.

Correct answer: B. (2, 7, 12)

In an arithmetic progression, the first term is \(a=2\) and the common difference is \(d=5\). Add 5 successively: \(2\), \(2+5=7\), and \(7+5=12\). Therefore, the correct sequence is (2, 7, 12). Option A does not maintain a common difference of 5. Exam tip: start with the first term and add the common difference to obtain each next term.

Correct answer: A. (9,11,13,15,\ldots)

An arithmetic progression is determined by two pieces of information: its first term \(a\) and its common difference \(d\). Starting with \(a=9\), add \(d=2\) repeatedly. This gives the pattern \(9, 9+2, 9+2+2, 9+2+2+2,\ldots\). Equivalently, its general term is \(a_n=a+(n-1)d\), so here \(a_n=9+2(n-1)\).

The first listed sequence is \(9,11,13,15,\ldots\). It begins with 9, and each consecutive difference is 2: \(11-9=2\), \(13-11=2\), and \(15-13=2\). Therefore choice A matches both requirements. Option C begins with 9 but has common difference 9, while option B begins with 2 and option D begins with 11, so none of those satisfies both conditions.

Correct answer: C. (20, 15)

The difference between consecutive terms is constant: \(35-40=-5\), \(30-35=-5\), and \(25-30=-5\). Hence, this is an AP with common difference \(-5\). Therefore, the terms after 25 are \(25-5=20\) and \(20-5=15\). In option (22, 19), the difference is \(-3\), so it does not continue the AP. Exam tip: Find the difference between consecutive terms first, then apply the same difference to get the next terms.

Correct answer: D. 30

The explicitly listed terms are 18, 21, 24 and 27. The common difference is \(21-18=3\), so the next term would be \(27+3=30\). Therefore, 30 is a term of the AP but is not among the terms given in the question. Exam tip: distinguish the terms already listed from the terms that can be generated next.

Correct answer: C. Constant-term

The common difference of an arithmetic progression is the difference between consecutive terms. Here, \(d=5-5=0\), and the difference between every pair of consecutive terms is also zero. Therefore, all the terms are equal, so it is an arithmetic progression with constant terms. It is neither increasing nor decreasing. Exam tip: an AP with \(d=0\) is called a constant AP.

Correct answer: B. 2

In an arithmetic progression, the common difference is the difference between two consecutive terms. Here, \(14-12=2\) and \(16-14=2\), so the common difference is 2. Option 4 is incorrect because it is the difference between the first and third terms, not consecutive terms. Exam tip: subtract each term from the term immediately after it to find the common difference.

Correct answer: C. 5

Find the differences between consecutive terms: \(45-50=-5\) and \(40-45=-5\). Thus, each term is 5 less than the preceding term, and the common difference is \(-5\). Since the question asks how much less, the answer is 5 rather than \(-5\). Exam tip: “how much less” gives the positive magnitude of the decrease, whereas the common difference is negative.

Correct answer: B. 1/2

The defining property of an arithmetic progression is that the difference between successive terms is constant. Since all terms here have denominator 4, subtract the numerators while keeping the denominator: 5/4 - 3/4 = 2/4 = 1/2. Check the next pairs as well: 7/4 - 5/4 = 2/4 = 1/2 and 9/4 - 7/4 = 2/4 = 1/2. Thus the common difference is 1/2, so option B is correct. Option A would result from incorrectly treating the numerator increase as 1/4. Option C is the first term, not the difference, and option D is not obtained from subtracting consecutive terms. Simplifying 2/4 to 1/2 is necessary for the standard answer.

Correct answer: C. -6

In an arithmetic progression, the common difference is found by subtracting the first term from the second term: d = a₂ − a₁ = 10 − 16 = −6. Therefore, option C is correct. Option B gives only the magnitude, 6, and misses the negative sign. Exam tip: when the next term is smaller than the previous term, the common difference is negative.

Correct answer: A. Arithmetic progression

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.

Correct answer: B. They increase

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.

Correct answer: C. They decrease

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 \(d<0\).

Correct answer: B. Second term

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.

Correct answer: B. -3

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.

Correct answer: B. 5

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.

Correct answer: B. \(8,8,8,8,\ldots\)

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.

Correct answer: A. Arithmetic progression

The governing definition is that a sequence is an arithmetic progression when the difference between every term and the term immediately after it remains constant. This fixed value is called the common difference, usually written as d. For example, in 4, 7, 10, 13, the successive differences are 3, 3, and 3, so the sequence is an arithmetic progression. A geometric progression instead has a constant ratio, as in 2, 6, 18, where the ratio is 3. A set is a collection of objects and need not have an order or equal differences, while an equation states that two expressions are equal. Therefore, option A is the correct name.

Correct answer: A. \(2, 5, 8, 11, \ldots\)

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.

Correct answer: B. -2

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.