Introduction to APs and common difference.

Correct answer: A. a+2d

The terms of an arithmetic progression are obtained by repeatedly adding the common difference d to the first term a: the first term is a, the second is a+d, and the third is a+2d. Therefore, the correct answer is a+2d. Remember that d is added twice for the third term; it is not multiplied by a.

Correct answer: B. 6

The difference between consecutive terms is constant: 4.5−4=0.5, 5−4.5=0.5, and 5.5−5=0.5. Thus, the common difference is 0.5, so the next term is 5.5+0.5=6. Option A is incorrect because it adds only 0.25. In an exam, first find the common difference and then add it to the last given term.

Correct answer: B. It is an arithmetic progression and (d=0)

An arithmetic progression is a sequence in which the difference between consecutive terms remains constant. In the sequence \(3,3,3,3,\ldots\), subtracting one term from the next gives \(3-3=0\) every time. Therefore the common difference is \(d=0\). The fact that the terms do not increase or decrease does not disqualify the sequence from being an arithmetic progression.

The first term is \(a=3\), not zero. Thus the correct statement is option B: it is an arithmetic progression with common difference zero. A constant sequence is an important special case of an arithmetic progression. The value 3 is the repeated term, while 0 describes the change between successive terms.

Correct answer: C. -3

The common difference of an arithmetic progression is the difference between two consecutive terms. Thus, \(d=11-14=-3\). This is verified by \(8-11=-3\) and \(5-8=-3\). Therefore, -3 is correct. Option A results from missing the negative sign. Exam tip: calculate the common difference as the second term minus the first term.

Correct answer: D. Arithmetic progression

The direct answer is option D: an arithmetic progression. An arithmetic progression, or AP, is a sequence in which the difference between every pair of consecutive terms is the same. The heights are 10, 16, 22, 28, and so on. Calculate the consecutive differences step by step: \(16-10=6\), \(22-16=6\), and \(28-22=6\). Since the common difference is always 6 cm, the sequence is an AP with first term \(a=10\) and common difference \(d=6\). Option D is correct. Option A, a geometric progression, would require a constant ratio, not a constant difference. Here the ratios are \(16/10\), \(22/16\), and \(28/22\), which are not equal. Option B, a simple list, is not the best mathematical classification because the terms follow a clear fixed-difference rule. Option C, no sequence, is wrong because a sequence is simply an ordered list of terms, and these heights are clearly ordered. The ladder story tells us exactly why the difference is constant: each next rung is 6 cm higher than the previous one. A positive common difference means the AP increases. Do not confuse equal addition with equal multiplication. Memory cue: AP means add the same number each time; GP means multiply by the same number.

Correct answer: A. 50 रुपये

In an arithmetic progression formed by repeated weekly savings increases, the common difference is the fixed amount added from one week to the next. Subtract the first two listed savings amounts: 150 − 100 = 50 rupees. The same result appears in the following steps: 200 − 150 = 50 and 250 − 200 = 50. Thus the sequence has common difference d = 50 rupees, so option A is correct. The amount 100 is the initial saving, not the weekly increase. The values 150 and 250 are later terms of the progression, so they cannot represent its common difference. The real-life wording does not change the AP rule.

Correct answer: D. 5

The common difference is the difference between two consecutive terms. Here, \(45-40=5\), \(50-45=5\), and \(55-50=5\). Therefore, the common difference of the ticket prices is 5. Option 10 is incorrect because it is not the difference between consecutive terms. Exam tip: To find the common difference in an AP, subtract the first term from the second term.

Correct answer: A. 4

The second term is 10 and the third term is 14. Therefore, their difference is 14 − 10 = 4. This is an arithmetic progression, so the difference between consecutive terms is constant. Exam tip: subtract the earlier term from the later term when finding the difference between consecutive terms.

Correct answer: B. (18, 14, 10, 6, …)

In an arithmetic progression, the difference between every pair of consecutive terms is constant. In option B, 14−18=−4, 10−14=−4, and 6−10=−4, so its common difference is d=−4. Option A has a difference of +4, while the differences in C and D are not constant. An easy exam check is to subtract each term from the term immediately before it.

Correct answer: C. 5

The common difference is found by subtracting one term from the next consecutive term. Here,
\((x+5)-x=5\), so the common difference is 5. Option D, \(x+5\), is a term of the sequence, not the common difference. Exam tip: In an AP, the difference between every pair of consecutive terms remains constant.

Correct answer: B. 3

The common difference of an arithmetic progression is the difference between two consecutive terms. Here, \(d=(2p+3)-2p=3\), and the next difference is also \((2p+6)-(2p+3)=3\). Therefore, the correct answer is 3; 6 is the total increase from the first to the third term, not the common difference. Exam tip: subtract the first term from the second term to find the common difference.

Correct answer: D. (2,4,7,11,\ldots)

In (2,4,7,11,\ldots), the differences are (2,3,4). To identify an arithmetic progression all differences must be equal.

Correct answer: A. 2

First identify the common difference by subtracting consecutive terms. We have −7 − (−10) = 3, −4 − (−7) = 3, and −1 − (−4) = 3. Therefore this is an arithmetic progression with d = 3. To obtain the next term, add the same difference to the last displayed term: −1 + 3 = 2. Hence option A is correct. Option B would result from adding 2 instead of the established difference; option C moves in the wrong direction and ignores that the sequence is increasing; and option D adds 4. Careful handling of the negative sign is important: subtracting a negative number is equivalent to adding its positive value.

Correct answer: B. 5

In an arithmetic progression, the second term is found by adding the common difference to the first term: second term = first term + d = 1 + 4 = 5. Therefore, 5 is correct. Option 4 is only the common difference, not the second term. Exam tip: In an AP, the difference between consecutive terms is always d.

Correct answer: C. 5

To find the equal increase, subtract each term from the term immediately after it: \(30-25=5\) and \(35-30=5\). Therefore, the common difference of this sequence is \(d=5\). Option 10 is incorrect because it is not the difference between consecutive terms. Exam tip: In an AP, use \(d=a_{n+1}-a_n\).

Correct answer: B. 4

Each successive term is 4 less than the preceding term: 12−16=−4, 8−12=−4, and 4−8=−4. Therefore, the magnitude of the equal decrease is 4. Note that the common difference of the AP is −4, whereas the decrease asked here is 4. Exam tip: subtract consecutive terms to verify the equal decrease.

Correct answer: C. (9, 9, 9)

The nth term of an arithmetic progression is \(a_n=a+(n-1)d\). Here, \(a=9\) and \(d=0\), so the second term is \(9+0=9\) and the third term is \(9+2\times0=9\). Therefore, the first three terms are \((9,9,9)\). Option B would represent a common difference of \(-9\), not 0. Exam tip: when the common difference is zero, every term of the AP equals the first term.

Correct answer: A. Arithmetic progression

Each consecutive time has the same difference of (8), so it is an arithmetic progression. In word problems identify the equal difference.

Correct answer: B. (11-7=4)

The direct answer is option B, \(11-7=4\). In an arithmetic progression, the common difference is found by subtracting the previous term from the next term: \(d=a_2-a_1\). Here the first term is 7 and the second is 11, so \(d=11-7=4\). Checking further gives \(15-11=4\) and \(19-15=4\). Option A reverses the subtraction and gives -4, not the forward common difference. Option B uses the correct order and is correct. Option C adds the terms, but addition does not measure the gap between terms. Option D multiplies them, which is also unrelated to common difference. Memory cue: next term minus previous term.

Correct answer: B. (4)

The common difference in an arithmetic progression is the change from one term to the next. It is calculated by subtracting an earlier term from the following term. If the sequence is truly arithmetic, this result stays the same for all consecutive pairs. This idea works equally well for increasing and decreasing sequences, and the difference may be positive, zero, or negative.

For the given sequence, subtract the first term from the second: \(8-4=4\). The other differences confirm it: \(12-8=4\) and \(16-12=4\). Hence the common difference is \(4\), making choice B correct. The number 8 is a term of the sequence, but it is not the common difference; this distinction is important. Values 2 and 12 also do not represent the change between consecutive terms.

Correct answer: B. -3

The common difference of an arithmetic progression is the difference between two consecutive terms: second term − first term. Here, \(d=12-15=-3\), and the same value is obtained from \(9-12=-3\) and \(6-9=-3\). Therefore, option B, \(-3\), is correct. Since the progression is decreasing, its common difference is negative; writing only its magnitude, \(3\), would be incorrect. Exam tip: calculate \(a_2-a_1\) or subtract any term from the term immediately following it.

Correct answer: C. 0

For an arithmetic progression, the common difference is obtained by subtracting one term from the next, for example d = a₂ − a₁. In this sequence the first two terms are both 9, so d = 9 − 9 = 0. The same result is obtained from every consecutive pair because each term equals the next term. A constant sequence is therefore a valid arithmetic progression with zero common difference; an AP does not have to increase or decrease. Option A confuses a term with the difference. Option B would apply to a sequence that increases by one each time, and option D would represent a decrease of nine at each step. Thus option C is correct.

Correct answer: A. 6

The first term of an arithmetic progression is the number written at the beginning of the sequence. In 6, 13, 20, 27, \ldots, the first term is therefore 6. The number 7 is the common difference, while 13 is the second term. Exam tip: identify the first term by locating the first number in the progression.

Correct answer: C. 23

This is an arithmetic progression with common difference \(11-5=6\). Therefore, the missing term is \(17+6=23\). Options 21 and 22 do not maintain a difference of 6 between consecutive terms. In an exam, find the common difference from two consecutive terms and add it to the preceding term.

Correct answer: A. 11

In an arithmetic progression, the second term is the sum of the first term and the common difference: \(a_2=a+d\). Thus, \(a_2=7+4=11\), so 11 is correct. The options 12 and 28 result from calculation or operation errors, while 3 comes from subtracting the common difference. Exam tip: for the second term, use \(a+d\) directly.