Introduction to APs and common difference.

Correct answer: C. (3, 3, 3)

In an arithmetic progression, the same common difference d is added successively. Here, a = 3 and d = 0, so the second term is 3 + 0 = 3 and the third term is also 3 + 0 = 3. Therefore, the first three terms are (3, 3, 3). Option D has common difference 3, not 0. Exam tip: When d = 0, every term of an AP equals its first term.

Correct answer: B. (15, 11, 7, 3)

In an AP, the first term is \(a=15\) and the common difference is \(d=-4\). Therefore, subtract 4 to get each next term: \(15,\;15-4=11,\;11-4=7,\;7-4=3\). Hence, the first four terms are \((15, 11, 7, 3)\). Option D starts from 11, so it omits the first term. Exam tip: a negative \(d\) means that the terms decrease.

Correct answer: B. \(3\)

To find the common difference, subtract a term from the term immediately after it. Here, \((2k+3)-2k=3\) and \((2k+6)-(2k+3)=3\). Thus, the difference between every pair of consecutive terms is \(3\). \(2k+3\) is the second term, not the common difference. Exam tip: In an AP, check the differences of at least two consecutive pairs of terms to confirm that they are equal.

Correct answer: B. 8.5

The second term of an AP is \(a_2=a_1+d\). Here, \(a_1=10\) and \(d=-1.5\), so \(a_2=10+(-1.5)=8.5\). The value 9.5 would result if the common difference were \(-0.5\), so it is not correct. Exam tip: adding a negative common difference decreases the term.

Correct answer: B. \( \frac{1}{3} \)

For an AP, \(d\) is the difference between any two consecutive terms. \( d=\frac{2}{3}-\frac{1}{3}=\frac{1}{3} \). Also, \( 1-\frac{2}{3}=\frac{1}{3} \), so \( d=\frac{1}{3} \).

Correct answer: B. 4

The difference between consecutive terms is the same: \(22-18=4\), \(26-22=4\), and \(30-26=4\). Hence, this is an arithmetic progression with common difference \(4\). Although \(2\) is a possible number, it is not the difference between any two consecutive terms here. Exam tip: To find the common difference, subtract the first term from the second term.

Correct answer: A. 36

The difference between consecutive terms is constant: \(57-64=-7\). Hence, this is an AP with common difference \(d=-7\). Since the fourth term is 43, the fifth term is \(43-7=36\). Option 37 would result from incorrectly using a common difference of \(-6\). Exam tip: To find the next term of an AP, add the common difference to the preceding term.

Correct answer: B. \(9\)

Given \(a=4\) and \(a+d=13\). Thus, \(4+d=13\), so \(d=13-4=9\). In an AP, \(d\) is the common difference. \(13\) is the second term here, not the common difference. Exam tip: when \(a+d\) is given, subtract \(a\) from it to find \(d\).

Correct answer: B. (0.25,0.5,0.75,1.0,\ldots)

In option B, the difference between consecutive terms is constant: \(0.5-0.25=0.25\), \(0.75-0.5=0.25\), and \(1.0-0.75=0.25\). Therefore, it is an AP with \(d=0.25\). Option A has common difference \(0.5\), while option C does not have a constant difference. Exam tip: To identify an AP, check the difference between every pair of consecutive terms.

Correct answer: B. 0.5

The common difference of an AP is found by subtracting a term from the term immediately after it. Here, \(7.5-7=0.5\) and \(8-7.5=0.5\), so the common difference is \(0.5\). The value \(1\) is not the difference between consecutive terms. In exams, verify the difference using any two consecutive terms.

Correct answer: B. 3

The second term of the sequence is 4 and the third term is 7. Hence, the difference between the third and second terms is \(7-4=3\). Therefore, 3 is correct. The number 4 is the second term, not the difference. Exam tip: To find the difference between consecutive terms, subtract the earlier term from the later term.

Correct answer: B. \(3\)

The common difference of an AP is found by subtracting a term from the term immediately after it. Here, \((a+1)-(a-2)=a+1-a+2=3\). Also, \((a+4)-(a+1)=3\), so the common difference is \(3\). \(a+3\) could represent a term, not the difference. Exam tip: When subtracting algebraic expressions, use brackets carefully to handle signs correctly.

Correct answer: D. (2,3,5,8,\ldots)

In the sequence (2,3,5,8,\ldots), the differences between successive terms are \(3-2=1\), \(5-3=2\), and \(8-5=3\). Since these differences are not equal, it is not an arithmetic progression. In contrast, the common differences in A, B, and C are \(5\), \(-4\), and \(0\), respectively. Exam tip: subtract consecutive terms; a sequence is an AP only when this difference remains constant.

Correct answer: B. 7

In an arithmetic progression, the common difference is d = a₂ − a₁. Here, a₁ = 11 and a₂ = 18, so d = 18 − 11 = 7. Therefore, the correct answer is 7. The value 6 is incorrect because it is not the difference between 11 and 18. Exam tip: To find the common difference, subtract an earlier term from the next term.

Correct answer: A. (2.5, 4.0, 5.5)

In an AP, the first term is a, and the common difference d is added to obtain each next term. Here the first three terms are 2.5, 2.5 + 1.5 = 4.0, and 4.0 + 1.5 = 5.5. Therefore, option A is correct. In option C, the first term is correct, but its common difference is 2.5, not 1.5. Exam tip: For the first three terms, use a, a + d, and a + 2d.

Correct answer: B. \(12\)

This is an arithmetic progression because \(6\) is added to each successive term. Here, the first term is \(a=-18\) and the common difference is \(d=6\). Therefore, \(a_6=a+5d=-18+5\times6=12\). Hence, the sixth term is \(12\). Option \(6\) is the fifth term, so it is a close but incorrect distractor. Exam tip: while finding the \(n\)th term, use \(a_n=a+(n-1)d\) carefully.

Correct answer: B. -9

In an AP, the first term is \(a=28\) and the second term is \(a+d=19\). Therefore, \(d=19-28=-9\). Hence, the correct answer is \(-9\). Choosing \(9\) misses the negative sign of the difference. Exam tip: To find the common difference, subtract the previous term from the next term.

Correct answer: B. (-6)

For an A.P., the common difference \(d\) is the constant value of \(\text{next term} - \text{previous term}\). Using the first two terms: \[d=(3m-6)-(3m)=-6.\] The \(3m\) cancels, leaving a constant difference \(-6\). Therefore option B is correct.

Correct answer: C. (16)

The difference from (23) to (30) is (7), so (k=23-7=16). For a missing term use the same difference backward.

Correct answer: A. \(-8,\ 5\)

In an AP, \(a\) is the first term, so \(a=-8\). The common difference is the difference between consecutive terms: \(d=-3-(-8)=5\). Therefore, \((a,d)=(-8,5)\). The option \((-8,-5)\) results from an incorrect sign while subtracting a negative number. Exam tip: write \(d=a_2-a_1\) and use brackets around negative terms.

Correct answer: C. Because consecutive differences are not equal

An arithmetic progression is defined by a constant difference between every pair of consecutive terms. For this sequence, the differences are 7 − 3 = 4, 13 − 7 = 6, and 21 − 13 = 8. Since 4, 6, and 8 are not equal, the sequence does not satisfy the defining condition. Therefore option C is correct. The first term may be any number, and positive terms can certainly form an arithmetic progression, so A and B are irrelevant. An infinite progression does not need a final term, so D is also incorrect. The terms increase, but mere increase is not enough; the amount of increase must remain constant.

Correct answer: A. \(7\)

The \(n\)th term of an AP is \(a_n=a+(n-1)d\). So the third term is \(a_3=a+2d=15+2(-4)=15-8=7\).

Correct answer: C. 3

To find the common difference, subtract the first term from the second term: \((4x+3)-4x=3\). Checking the next pair, \((4x+6)-(4x+3)=3\) as well. Therefore, the common difference is 3. The number 6 is part of the third term, not the common difference. Exam tip: In an AP, the difference between every pair of consecutive terms must be the same.

Correct answer: B. \(-5\)

In an arithmetic progression, the common difference is \(d=a_2-a_1\). Here, \(d=51-56=-5\), so each successive term is 5 less than the preceding term. \(5\) is only the magnitude of the decrease; the common difference must be negative. Exam tip: always subtract the first of two consecutive terms from the second.

Correct answer: C. 14

For three consecutive terms of an arithmetic progression, the middle term is the average of the first and third terms. Thus, \(m=\frac{8+20}{2}=14\). Therefore, 14 is correct. If 12 or 15 is chosen, the common differences on the two sides will not be equal. Exam tip: for consecutive AP terms \(a,b,c\), use \(2b=a+c\).