Introduction to APs and common difference.

Correct answer: B. \((-4,2,8,14,\ldots)\)

In an AP, \(a\) is the first term and \(d\) is the difference between consecutive terms. In option B, the first term is \(-4\), and \(2-(-4)=6\) and \(8-2=6\). Hence, it has \(a=-4\) and \(d=6\). Option C also has common difference 6, but its first term is 4. Exam tip: check the first term first, then subtract the first term from the second term to find \(d\).

Correct answer: A. (3)

The common difference is (\frac{1}{2}), so the next term is (\frac{5}{2}+\frac{1}{2}=3). Keep denominators common when adding fractions.

Correct answer: C. \(s\)

The common difference of an AP is the difference between consecutive terms. Here, \((5+s)-5=s\) and \((5+2s)-(5+s)=s\), so the common difference is \(s\). \(2s\) is the difference between the first and third terms, not between consecutive terms. Exam tip: Find the common difference by subtracting the first term from the second term.

Correct answer: B. (20)

The equal hourly increase is (20) rupees, so (d=20). In word problems treat the equal increase as the common difference.

Correct answer: A. \(20,17,14,11,\ldots\)

In the sequence \(20,17,14,11,\ldots\), the consecutive differences are \(17-20=-3\) and \(14-17=-3\). Therefore, its common difference is \(d=-3\), not \(-2\). In each of the other three sequences, every next term is 2 less, so \(d=-2\). Exam tip: find the common difference using \(d=a_2-a_1\).

Correct answer: D. 10

The nth term of an AP is \(a_n=a+(n-1)d\). Therefore, \(a_4=10+(4-1)\times 0=10\). When \(d=0\), every term is equal to the first term; a value such as 20 would occur only if the common difference were non-zero. Exam tip: Substitute the term number \(n\) and common difference \(d\) carefully in the formula.

Correct answer: C. 13

In this AP, the first term is \(a=3\) and the common difference is \(d=8-3=5\). Therefore, \(a+2d=3+2(5)=13\). Hence, 13 is the correct option. The number 8 is the second term, whereas \(a+2d\) represents the third term of an AP. Exam tip: use \(a+(n-1)d\) to find the \(n\)th term.

Correct answer: B. 3

In an arithmetic progression, the common difference \(d\) is the difference between consecutive terms. Here, \((y+1)-(y-2)=y+1-y+2=3\). Also, \((y+4)-(y+1)=3\), so \(d=3\). \(y-2\) and \(y+1\) are terms of the AP, not the common difference. Exam tip: Find \(d=a_2-a_1\) and verify it using the next pair of terms.

Correct answer: D. (30) and (-5)

Find the common difference from consecutive terms: d=45−50=−5, and the same change appears in 40−45 and 35−40. To obtain the next term, subtract 5 from the last displayed term: 35+(−5)=30. The question asks for the next term and d in that order, so the pair is (30,−5). Hence option D is correct; option A has the wrong sign for d.

Correct answer: A. (12,8,4)

In an arithmetic progression, the first term is a, and each next term is obtained by adding the common difference d. Here, a=12 and d=-4, so the terms are 12, 12+(-4)=8, and 8+(-4)=4. Therefore, the correct order is (12,8,4). Option B uses a common difference of +4, while option C does not begin with the first term 12. Exam tip: A negative d means that the same value is subtracted from each successive term.

Correct answer: C. \(a=0,\ d=-3\)

In an AP, \(a\) is the first term, so \(a=0\). The common difference \(d\) is the difference between consecutive terms: \(d=-3-0=-3\). Hence, \(a=0,\ d=-3\) is correct. In option A, the sign of the common difference is incorrect. Exam tip: use \(d=a_2-a_1\), subtracting the first term from the second term.

Correct answer: D. 30

The AP is 20, 35, 50, 65, \ldots, where \(a_1=20\) and \(a_3=50\). Therefore, \(a_3-a_1=50-20=30\). The common difference is \(d=15\), but the difference between the first and third terms is \(2d=30\), so 15 is not correct. Exam tip: When asked for the difference between two terms, subtract the stated terms directly to verify your answer.

Correct answer: B. (5)

Adding the same (2) to each term does not change the difference, so (d=5). Equal change in all terms does not change (d).

Correct answer: A. (3)

The common difference is (-2-(-7)=5), so the third term is (-2+5=3). Watch signs carefully with negative numbers.

Correct answer: D. \(-5\)

The common difference is found by subtracting a term from the next term: \((2a-5)-2a=-5\). Also, \((2a-10)-(2a-5)=-5\), confirming that the difference is constant. Each next term decreases by 5, so the common difference is \(-5\); \(5\) is only the magnitude of the decrease. Exam tip: always keep the sign while writing the common difference.

Correct answer: C. 15

For three consecutive terms of an arithmetic progression, the middle term equals the average of the first and third terms. Thus, \(b=\frac{9+21}{2}=15\). Therefore, 15 is the correct option. For instance, if \(b=13\), the differences are \(13-9=4\) and \(21-13=8\), which are not equal. Exam tip: For three consecutive AP terms \(a-d,\ a,\ a+d\), the middle term is always the average of the outer terms.

Correct answer: B. (\frac{1}{3},1,\frac{5}{3},\frac{7}{3},\ldots)

In (\frac{1}{3},1,\frac{5}{3},\frac{7}{3},\ldots), every difference is (\frac{2}{3}). Check fractions using common denominators.

Correct answer: A. 4, 9, 14, 19, ...

In option A, consecutive differences are equal: 9 − 4 = 5, 14 − 9 = 5, and 19 − 14 = 5. Hence it is an AP. The differences change in the other options. Exam tip: check at least two consecutive differences.

Correct answer: B. 0.8

In an arithmetic progression, the common difference \(d\) is the difference between consecutive terms. Here, \(d=6.8-6=0.8\). Checking further, \(7.6-6.8=0.8\) and \(8.4-7.6=0.8\). The value 0.6 would result from incorrect decimal subtraction. Exam tip: find \(d=a_2-a_1\) and verify it using the next pair of terms.

Correct answer: C. (10-15=-5)

The common difference is next term minus previous term, so (10-15=-5). In a decreasing sequence the order of subtraction is very important.

Correct answer: A. 7

In an arithmetic progression, the second term is obtained by adding the common difference to the first term: \(a_2=a_1+d\). Thus, \(11=a_1+4\), so \(a_1=7\). Option 11 is the second term, not the first term. Exam tip: subtract \(d\) from the second term to find the first term.

Correct answer: B. (12)

The difference from (18) to (24) is (6), so (u=18-6=12). While finding a missing term use the same difference forward and backward.

Correct answer: B. 5

The common difference is the difference between consecutive terms. Here, \(13-8=5\) and \(18-13=5\), so 5 is added to each term to obtain the next term. Therefore, the common difference is 5. Although 4 is a close distractor, it is not the difference between any two consecutive terms here. Exam tip: For an AP, quickly verify using \(d=a_2-a_1\).

Correct answer: B. -6

The common difference is found by subtracting a term from the next term: \(36-42=-6\). Also, \(30-36=-6\) and \(24-30=-6\), so the common difference is \(-6\). Option 6 ignores the sign; in a decreasing AP, the common difference is negative. Exam tip: always calculate \(a_2-a_1\) in this order to get the correct sign.

Correct answer: C. 18

This is an arithmetic progression because the differences 10-6 and 14-10 are both 4. Therefore, the term after 14 is 14+4=18. Checking further, 18+4=22. Option 16 would give a difference of 2 from 14, so it is not correct. Exam tip: In an AP missing-term question, verify that the common difference is the same on both sides of the blank.