Introduction to APs and common difference.

The governing concept is the common difference of an arithmetic progression. For option A, subtract consecutive terms: 6−2=4, 10−6=4, and 14−10=4. Thus its first three consecutive differences are exactly (4,4,4), and it is an arithmetic progression with common difference 4. In option B the differences are 4, 5, and 6, so they change. In option C each difference is −2, not +4. In option D the differences are 3, 5, and 7, which also are not constant. Hence only option A satisfies the stated condition. The repeated wording in the supplied option has been normalized, without changing the mathematical choices or answer.

In an arithmetic progression, the common difference d is found by subtracting one term from the next consecutive term. Using the first two terms gives d = 29 - 24 = 5. The other pairs confirm the same result: 34 - 29 = 5 and 39 - 34 = 5. Therefore the sequence increases by 5 at every step, and option B is correct. Option A is one less than the actual difference, option C does not arise from any consecutive subtraction in the displayed sequence, and option D is the difference between neither adjacent pair. The calculation must use consecutive terms rather than the first and last terms.

Answer: 11, so option C is correct. The defining property of an arithmetic progression is a constant difference between consecutive terms. From the known terms, 25 - 18 = 7 and 32 - 25 = 7. Therefore the common difference is 7. Since p appears immediately before 18, adding 7 to p must give 18. Thus p + 7 = 18, so p = 11. The resulting sequence is 11, 18, 25, 32, ... , and its consecutive differences are all 7. Option A, 9, would give 18 - 9 = 9; option B, 10, would give a difference of 8; and option D, 12, would give a difference of 6. Each would break the constant-difference rule. Memory cue: when a missing term is before a known AP term, subtract the common difference from that known term.

The governing rule of an arithmetic progression is that each new term is obtained by adding the same common difference to the preceding term. Here, 19 − 12 = 7, 26 − 19 = 7, and 33 − 26 = 7, so the common difference is consistently 7. To find the term after 33, add this difference once more: 33 + 7 = 40. Thus option C is correct. Option A would add 5, option B would add 6, and option D would add 8. Each of those choices breaks the established pattern. The ellipsis means that the same rule continues beyond the displayed terms; it does not signal a change in the operation after 33.

For an arithmetic progression, the common difference d is the same subtraction between every pair of consecutive terms. Here, 64−70=−6, 58−64=−6, and 52−58=−6, so d=−6. To obtain the next term, add this difference to the last known term: 52+(−6)=46. Therefore the next term and d, in the requested order, are 46 and −6, so option A is correct. Option B has the wrong sign for the difference and would make the sequence increase. Options C and D repeat terms already present rather than giving the next term. The negative difference correctly reflects that the sequence decreases by 6 each time.

The governing concept is the general form of an arithmetic progression. In three consecutive terms written as a, a+d, a+2d, the constant d is the common difference between successive terms. Comparing the given first two terms, 4+d=1, so subtracting 4 from both sides gives d=−3. This result is confirmed by the third term: 4+2d=4+2(−3)=4−6=−2, exactly as required. Therefore option B is correct. Option A has the wrong sign and would produce the sequence 4,7,10. Option C would make the second term 9, and option D would make the second term −1, so neither matches the supplied terms. The negative value simply means that the arithmetic progression decreases by 3 at each step.

The defining condition of an arithmetic progression is that the difference between every pair of consecutive terms must be the same constant. Calculate the successive differences: 5-2=3, 10-5=5, and 17-10=7. Since 3, 5, and 7 are unequal, no single common difference exists, so the sequence is not an arithmetic progression. Option C states this exact reason and is correct. Merely increasing terms do not guarantee an AP; for example, their gaps may change, as they do here. Positive differences alone are also insufficient, so option B is false. The size of the first term has no bearing on the definition, making option D irrelevant.

An arithmetic progression has the same difference between every pair of consecutive terms. In option A, 6-6=0 for every step, so its common difference is 0. Option B has common difference 6, and option C has common difference -6. Option D has differences 0, 6 and 0, which are not equal. Hence the constant sequence in option A is correct.

The governing property is that in a three-term arithmetic progression, the middle term is the arithmetic mean of the first and third terms. Therefore, b=(a+c)/2=(-6+18)/2=12/2=6. The requested expression is not b itself; it is b-a. Substituting the values gives b-a=6-(-6)=6+6=12. Hence option C is correct. Option A is the value of b, not b-a. Option B and option D do not satisfy the required subtraction and may result from using the endpoints incorrectly or confusing the common difference with the requested expression. The common difference is also 12, which agrees with the answer here because the first three terms are -6, 6, 18.

The expression (5, 5+d, 5+2d) represents three consecutive terms of an arithmetic progression whose first term is 5 and common difference is d. Match the second general term with the second given term: 5+d=-1. Subtracting 5 from both sides gives d=-1-5=-6. We can verify the result using the third term: 5+2d=5+2(-6)=5-12=-7, exactly as required. Therefore option B is correct. Option A has the wrong sign, usually caused by reversing the subtraction. Option C is the magnitude of the change from 5 to -7, but it ignores the fact that two equal steps occur. Option D incorrectly doubles the common difference.

The defining property of an arithmetic progression is that subtracting each term from the next gives the same constant. Calculate the first difference: (3a + 2b) − 3a = 2b. The next differences confirm the pattern: (3a + 4b) − (3a + 2b) = 2b and (3a + 6b) − (3a + 4b) = 2b. Thus the common difference is 2b, so option C is correct. The repeated expression 3a is a fixed part of every term and cancels whenever consecutive terms are subtracted; it is not the common difference. Similarly, b alone is not correct because the coefficient 2 is part of the change. The conclusion follows symbolically and does not require particular numerical values for a or b.

The first term is the initial number of a sequence, while the common difference is found by subtracting each term from the next. In option A, the first term is 6 and the differences are 12 − 6 = 6, 18 − 12 = 6, and 24 − 18 = 6. Hence a₁ = 6 and d = 6, so they are equal. In option B, d = 4, which is not equal to the first term. In option C, d = 0, and in option D, d = −6. Therefore only option A satisfies the required condition. The test must compare the first term with the repeated difference, not merely check whether the sequence is an AP, because all four listed sequences have a constant difference.

The common difference of an arithmetic progression is obtained by subtracting a term from the following term. Using the first two terms, d = 0.02 − 0.20 = −0.18. Verify the result with the next pairs: −0.16 − 0.02 = −0.18, and −0.34 − (−0.16) = −0.34 + 0.16 = −0.18. Since the difference remains constant, the sequence is an AP with common difference −0.18, so option A is correct. The negative sign shows that the terms decrease by 0.18 each time. Option B has the correct magnitude but wrong sign, option C comes from incorrect decimal subtraction, and option D is a term of the sequence rather than the difference.

For three consecutive terms of an arithmetic progression, the middle term is the arithmetic mean of the first and third, so a + c = 2b. Another useful representation is a = b − d and c = b + d. Adding all three terms gives a + b + c = (b − d) + b + (b + d) = 3b, because the opposite differences cancel. The question states that this sum is 27, so 3b = 27 and b = 9. Therefore option C is correct. It is not necessary to find a and c separately. The other choices do not satisfy the direct relation b = (a + b + c) / 3 for three consecutive AP terms, so they cannot be the middle term under the given condition.

The governing concept is the common difference of an arithmetic progression: subtract one term from the next and check whether the result is constant. For the first two terms, (2p + 2q) − (2p − q) = 2p + 2q − 2p + q = 3q. The next difference is (2p + 5q) − (2p + 2q) = 3q, and the following difference is (2p + 8q) − (2p + 5q) = 3q. Thus the sequence has a constant common difference, d = 3q, so option D is correct. The terms containing 2p cancel during subtraction; therefore 2p is not the difference. Options A and C are only parts of the change and do not match the consecutive-term calculation.

To test three symbolic terms for an arithmetic progression, calculate the two consecutive differences. The first is (a + b) − (a − 2b) = 3b. The second is (a + 4b) − (a + b) = 3b. These differences are identical for all values of a and b, so the three terms are always consecutive terms of an AP with common difference d = 3b. Therefore option A is correct. The quantity a cancels in both subtractions, so it cannot be the common difference. No extra condition such as a = b is required, and the equal differences disprove the claim that the terms are not in AP.

The governing concept is the common difference of an arithmetic progression. It is found by subtracting each term from the following term. Here, (x + √5) − x = √5. Checking the next pairs gives (x + 2√5) − (x + √5) = √5 and (x + 3√5) − (x + 2√5) = √5. Since the same quantity is added every time, the sequence is an AP with common difference √5. Therefore option B is correct. The symbol x is the initial term’s fixed part and disappears when consecutive terms are subtracted. Option C is twice the actual step, while option D is a complete term of the sequence rather than the difference between two terms. The result does not depend on the particular value of x.

The governing property is that consecutive terms of an AP have equal differences. Therefore, m − 14 must equal 50 − m. Solving gives 2m = 64, so m = 32. The common difference is then d = m − 14 = 32 − 14 = 18; it also agrees with 50 − 32 = 18. Hence option B is correct. Option A incorrectly uses 30 as the middle term and consequently gets a difference of 16. Options C and D place the middle term too high, so their two successive differences are not equal: 20 and 16 for C, and 22 and 14 for D. The arithmetic-mean rule, m = (14 + 50)/2, provides the same result.

The governing concept is the common difference of an arithmetic progression, defined as the difference between any term and the immediately preceding term. Subtract the first term from the second: d = 72 − 81 = −9. Checking the next pairs gives 63 − 72 = −9 and 54 − 63 = −9, confirming that the sequence is an AP with d = −9. Therefore option B is correct. The negative sign is essential because the terms decrease as the sequence advances. Option A gives the magnitude but loses the direction, option C doubles the actual change, and option D is merely a term of the sequence rather than a difference. Thus the constant signed difference is −9.

In an arithmetic progression, the common difference is the constant amount added to obtain each successive term. Subtracting consecutive terms gives 8 − 5 = 3, 11 − 8 = 3, and 14 − 11 = 3. Since the same value is obtained each time, the common difference is 3. Therefore, option B is correct; 2, 4, and 5 do not represent the change between consecutive terms.