Introduction to APs and common difference.

An arithmetic progression is a sequence in which the difference between every pair of consecutive terms remains constant. To find the common difference, subtract one term from the next: 7 − 3 = 4, 11 − 7 = 4, and 15 − 11 = 4. Since the same value is obtained each time, the sequence is an arithmetic progression with common difference d = 4. Therefore option C is correct. The number 7 is a term of the sequence, not the difference, while 2 and 3 do not equal the change between consecutive terms. Checking more than one pair confirms that the difference is constant.

In an arithmetic progression, the common difference is the difference between consecutive terms. If the first two terms are \(a_1\) and \(a_2\), then \(d=a_2-a_1\). Option C reverses the subtraction and gives \(-d\). In exams, subtract the earlier term from the later term.

The common difference is found by subtracting a term from the next term: \(d=8-10=-2\). The other consecutive differences are also \(6-8=-2\) and \(4-6=-2\), so the progression is decreasing with a negative common difference. Option A has the wrong sign. Exam tip: use \(d=a_{n+1}-a_n\), not the reverse subtraction.

In (5,10,15,20,\ldots), (5) is added each time. A sequence with equal differences is an arithmetic progression.

The difference of consecutive terms is (6-6=0). A constant sequence is also an arithmetic progression with common difference (0).

The first term of an arithmetic progression is the number with which the sequence begins. In the given progression, the first term is 12, so option B is correct. The number 17 is the second term, while 5 is the common difference, not the first term. Exam tip: To identify the first term, look at the leftmost number in the progression.

This is an arithmetic progression because the difference between consecutive terms is constant: \(9-4=5\) and \(14-9=5\). Hence, with common difference \(5\), the missing term is \(14+5=19\). The values 18 and 20 do not maintain the same common difference. Exam tip: first check the difference between consecutive known terms.

A sequence is an arithmetic progression only when the difference between consecutive terms is constant. For option A, the differences are 4, 4, and 4. For option B, they are -3, -3, and -3. For option C, they are 2, 2, and 2. However, option D gives 5 − 2 = 3, 9 − 5 = 4, and 14 − 9 = 5. Since these differences are not equal, option D is not an arithmetic progression. The changing increments distinguish it from the other three sequences, each of which has a fixed common difference.

In an arithmetic progression, the second term is the sum of the first term and the common difference: second term = a + d = 5 + 3 = 8. Therefore, option A is correct. Option D is only the first term, not the second term. Exam tip: remember the formula a₂ = a + d for finding the second term.

The common difference is the difference between two consecutive terms. Here, \(d=a_2-a_1=1-(-3)=4\); the other consecutive differences are also \(5-1=4\) and \(9-5=4\). Therefore, the correct answer is 4. The value 5 is a term, not the difference between consecutive terms. Exam tip: subtract any two consecutive terms to verify the common difference.

The common difference of an arithmetic progression is found by subtracting a term from the term immediately before it. Here, \\(20-25=-5\\) and \\(15-20=-5\\), so the common difference is \\(-5\\). Option A has the wrong sign; a decreasing progression has a negative common difference. Exam tip: subtract consecutive terms and always check the sign.

The common difference of an arithmetic progression is defined as the difference between a term and the term immediately before it: d=a₂−a₁. Here every new term is obtained by adding 7, so a₂−a₁=7 and the same calculation holds for every consecutive pair. Therefore d=7, making option C correct. Zero would mean no change, and −7 would describe a decreasing sequence.

The common difference of an arithmetic progression is the fixed amount added to obtain each term from the preceding term. It is found by subtracting one term from the next, such as \(a_2-a_1\). The terms may be whole numbers, fractions, or decimals; the method remains exactly the same. Every pair of consecutive terms should give the same difference.

For this sequence, subtract the first term from the second: \(2.0-1.5=0.5\). The next checks agree: \(2.5-2.0=0.5\) and \(3.0-2.5=0.5\). Thus the constant difference is \(0.5\), so choice B is correct. The value \(0.25\) would be too small, while \(1.0\) and \(1.5\) do not equal the change between consecutive terms.

The common difference of an arithmetic progression is found by subtracting any term from the term immediately after it. Using the first two terms, d = 1 − 1/2 = 1/2. Checking the next pairs confirms the result: 3/2 − 1 = 1/2 and 2 − 3/2 = 1/2. Thus the sequence increases by one-half at every step, so its common difference is 1/2. Option A is correct. The values 1, 3/2, and 2 are terms of the sequence, not the fixed amount added between consecutive terms. Working with fractions requires a common denominator, but here the subtraction is direct.

In option B, the differences between consecutive terms are equal: \(7-4=3\), \(10-7=3\), and \(13-10=3\). Therefore, it is an arithmetic progression with common difference \(d=3\). Option C has common difference 4, while option D has common difference −3. In an exam, find the common difference by subtracting each term from the term immediately after it.

In an arithmetic progression, subtract one term from the next to obtain the constant difference. Using the first two terms, d=(a+2)−a=2. The next checks give (a+4)−(a+2)=2 and (a+6)−(a+4)=2, confirming that the difference is constant. Therefore option C is correct; a and 2a are expressions involving the first term, not the term-to-term increase.

The common difference is the difference between consecutive terms: \(27-30=-3\), \(24-27=-3\), and \(21-24=-3\). Therefore, the next term is \(21+(-3)=18\). Option 19 is incorrect because it would give a difference of \(-2\), whereas the sequence decreases by 3 each time. Exam tip: identify the common difference first, then add it to the last given term.

Count the terms from left to right: the first term is 2, the second is 5, and the third is 8. The common difference is 3, but for this question, identifying the third position directly is sufficient. Therefore, the correct answer is 8; 5 is the second term and 11 is the fourth term. In an exam, do not confuse the term number with its value.

An arithmetic progression is a sequence in which the difference between consecutive terms remains constant. For example, in 3, 7, 11, 15, each subtraction gives 4, so d=4. The terms need not have equal squares or be zero. A rule in which each term is double the previous one describes a geometric progression, not an arithmetic progression. Hence option B states the defining property.

In an arithmetic progression, the difference between every pair of consecutive terms must be constant. Here, the differences are 6−3=3, 12−6=6, and 24−12=12. Since these are not equal, the sequence is not an arithmetic progression. In an exam, first check consecutive differences; being positive or integral does not make a sequence an AP.