Introduction to APs and common difference.

In an arithmetic progression, the first term is a and the common difference d is added to obtain each next term. Here, a = 11 and d = 6: 11, 11 + 6 = 17, 17 + 6 = 23, 23 + 6 = 29. Hence, the first four terms are (11, 17, 23, 29). In option A, 5 has been added instead of 6. Exam tip: Check that the difference between every pair of consecutive terms equals d.

Check the difference between consecutive terms: \(75-81=-6\), \(69-75=-6\), and \(63-69=-6\). Thus, this is an arithmetic progression with common difference \(-6\). Therefore, the next term is \(63-6=57\). Getting 60 would require subtracting 3, which does not follow the given common difference. Exam tip: For a next-term question, first find the difference between consecutive terms and apply it to the last term.

(\frac{6}{7}-\frac{4}{7}=\frac{2}{7}). For fractions with the same denominator, quickly check the difference of numerators.

The common difference of an arithmetic progression is obtained by subtracting the earlier term from the immediately following term. Since the consecutive terms are 32 and 41 in that order, d = 41 − 32 = 9. Therefore the progression increases by 9 at this step, and option C is correct. The value 73 comes from adding the two terms rather than finding their difference. Values 7 and 8 result from incorrect subtraction or estimation. No nth-term or sum formula is needed because two consecutive terms directly reveal the common difference. The same d would occur between every neighboring pair in a true arithmetic progression.

The common difference is (23-18=5), and the same difference continues. In exams always check the difference of consecutive terms.

In an arithmetic progression, the difference between every pair of consecutive terms is constant. The difference from 14 to 19 is 19 − 14 = 5, so the common difference is d = 5. Moving one step backward from 14 gives k = 14 − 5 = 9. This also checks with the beginning: k − 4 = 9 − 4 = 5. Thus the sequence is 4, 9, 14, 19, … and option D is correct. The other choices fail the equal-difference condition: they would not make both k − 4 and 14 − k equal to 5.

In an AP, \(a\) is the first term, so \(a=-2\). The common difference \(d\) is the difference between consecutive terms: \(d=3-(-2)=5\). Therefore, \(a=-2,\ d=5\) is correct. Option B incorrectly treats the second term, 3, as the first term. Exam tip: take extra care with signs when subtracting a negative number.

In an arithmetic progression, the difference between every pair of consecutive terms is constant. In option C, \(9-5=4\), \(13-9=4\), and \(17-13=4\). Hence, its common difference is \(4\), so it is an AP. In option B, the differences are \(3,5,7\), which are not equal. Exam tip: To identify an AP, quickly check consecutive differences.

The differences are (4,6,8), which are not equal. Increasing terms alone are not enough for an arithmetic progression.

In an AP, every consecutive difference must be equal. Here, 21-25=-4 and 17-21=-4, but 14-17=-3. Hence it is not an AP. Exam tip: check at least all adjacent differences, not merely whether terms increase or decrease.

The common difference is the amount added to one term to obtain the next term. With algebraic terms, the same rule is used: subtract the entire first term from the entire second term. Any variable parts that are identical cancel during subtraction, leaving the constant change between the terms.

Here the first term is \\(5x\\), and the second term is \\(5x+2\\). Thus, \\(d=(5x+2)-5x=2\\). The value of \\(x\\) does not matter because the two \\(5x\\) parts cancel. The third term also confirms the result: \\((5x+4)-(5x+2)=2\\). Therefore the sequence has common difference 2, so choice A is correct. Expressions such as \\(5x\\) or \\(5x+2\\) are terms, not the difference.

The governing concept is that consecutive terms of an arithmetic progression differ by a constant common difference d. From the known terms, d = 15 - 11 = 4, and this is confirmed because 19 - 15 = 4. The term before 11 must therefore be obtained by moving one step backward and subtracting the same difference: p = 11 - 4 = 7. Thus option C is correct. Option A would result from subtracting 8, option B from subtracting 6, and option D from subtracting only 2; none preserves the constant difference of 4 throughout the sequence.

In an arithmetic progression, the common difference is d = second term − first term. Here, d = 27 − 31 = -4. Each successive term is 4 less than the preceding term, so the common difference is negative. Although 4 is the magnitude of the decrease, it is not the common difference. Exam tip: always subtract an earlier term from the next consecutive term to find d.

The middle term is the average of (3) and (11), so (m=\frac{3+11}{2}=7). In three consecutive terms the middle term is the average.

In option B, the differences between consecutive terms are \(34-40=-6\), \(28-34=-6\), and \(22-28=-6\). Hence, it is an arithmetic progression with \(d=-6\). In option C, the differences are \(-6,-4,-6\), so they are not constant. Exam tip: To identify an AP, check the difference between every pair of consecutive terms.

(1-\frac{3}{4}=\frac{1}{4}) and (\frac{5}{4}-1=\frac{1}{4}). For fractions subtract using common denominators.

In an arithmetic progression, the common difference d is added to each term to obtain the next term. Here, a = -5 and d = 4: -5, -5 + 4 = -1, -1 + 4 = 3, and 3 + 4 = 7. Therefore, the first four terms are (-5, -1, 3, 7). In option A, the common difference is 3, not 4. Exam tip: Always check the difference between consecutive terms.

To find the common difference, subtract a term from the term immediately after it: \(0.5-0.2=0.3\). Checking further, \(0.8-0.5=0.3\), so the common difference is 0.3. The number 0.2 is the first term, not the common difference. Exam tip: find the difference between consecutive terms and verify it with another pair.

The defining rule for an arithmetic progression is d = a next term − a preceding term. Since the consecutive terms are 12 followed by 17, calculate d = 17 − 12 = 5. Thus option C is correct. The order matters: subtracting the earlier term from the later term gives the forward common difference. Option A is the sum 12 + 17, not the difference; option B is merely the first supplied term; and option D is the second supplied term. Because the terms are consecutive, no formula for the sum or nth term is needed. The same difference would continue in the progression, so the next term after 17 would be 22.

In an arithmetic progression, the same number is added or subtracted to get each next term. That fixed number is called the common difference. In the given sequence, the terms increase regularly, so we can find the next term by adding the difference to the last displayed term. This makes option B the correct answer.

The consecutive differences are \(13-8=5\), \(18-13=5\), and \(23-18=5\). Therefore the common difference is 5. Adding it to the last term gives \(23+5=28\). Hence the next term is 28. The other choices do not continue the established pattern because they add 3, 7, or 8 instead of 5.