Arithmetic Progression

Correct answer: B. (3, 10, 17, 24)

Explanation: In an arithmetic progression, the first term is 3 and 7 is added to obtain each next term: 3, 3+7=10, 10+7=17, 17+7=24. Therefore, the correct sequence is (3, 10, 17, 24). In option C, the first term has incorrectly been changed to 7. Exam tip: write the first term first, then add the common difference successively.

Correct answer: B. (a_n=30-5n)

Explanation: At (n=1) it gives (25), and at (n=2) it gives (20), so (a_n=30-5n). In exams, check a decreasing rule with the first term.

Correct answer: B. 10

Explanation: Given \(a_n=30-5n\). Substituting \(n=4\), we get \(a_4=30-5(4)=30-20=10\). Therefore, 10 is the correct option. Getting 15 would be incorrect because \(5\times4=20\), not 15. Exam tip: To find a particular term, substitute the given value of \(n\) carefully in the formula.

Correct answer: A. \(a=2,\ d=7\)

Explanation: In an arithmetic progression, the first term is denoted by \(a\), so \(a=2\). The common difference \(d\) is the difference between consecutive terms: \(d=9-2=7\). Therefore, \(a=2,\ d=7\) is correct. In \(a=7,\ d=2\), the first term and common difference have been interchanged. Exam tip: find \(d\) by subtracting an earlier term from the next consecutive term.

Correct answer: C. 22

Explanation: The nth term of an arithmetic progression is \(a_n=a+(n-1)d\). Thus, \(a_5=6+(5-1)\times4=6+16=22\). Therefore, 22 is correct. The terms are 6, 10, 14, 18, 22, so 18 is the fourth term, not the fifth. Exam tip: To find the fifth term, add the common difference four times to the first term.

Correct answer: C. (6)th term

Explanation: The terms are (4,7,10,13,16,19), so (19) is the sixth term. In exams, include the first term while counting.

Correct answer: C. 6

Explanation: In an arithmetic progression, the common difference is the difference between consecutive terms. Thus, \(d=a_2-a_1=14-8=6\). Therefore, 6 is the correct option. Choosing 5 does not give the correct subtraction of the first term from the second term. Exam tip: When the first two terms are given, use \(d=a_2-a_1\).

Correct answer: C. 33

Explanation: The governing concept is the common difference of an arithmetic progression. Subtract consecutive terms: 21 − 17 = 4, 25 − 21 = 4, and 29 − 25 = 4. Since the difference remains constant, the next term is found by adding 4 to the last displayed term: 29 + 4 = 33. Thus option C is correct. This is not a geometric progression, because the ratios are not constant; the question requires addition of the common difference, not multiplication by a common ratio. Option A is obtained by adding only 2, option B by adding 3, and option D by adding 5. None of those changes agrees with the repeated difference of 4 observed in every pair of consecutive terms.

Correct answer: C. 5, 10, 15, 20

Explanation: An arithmetic progression is a sequence in which the difference between consecutive terms remains constant. In option C, 10 − 5 = 5, 15 − 10 = 5, and 20 − 15 = 5, so the common difference is 5. The differences in A double, those in B are 2, 3, and 4, and those in D are 3, 5, and 7. Therefore, only option C is an arithmetic progression.

Correct answer: B. (60)

Explanation: The direct answer is option B, 60. The sequence decreases by 10 each time: \(90-100=-10\), \(80-90=-10\), and \(70-80=-10\). Starting from the first term, the terms are first 100, second 90, third 80, fourth 70, and fifth \(70-10=60\). Hence option B is correct. Option A, 50, would be the sixth term, not the fifth. Option C, 70, is already the fourth term. Option D, 80, is already the third term. The negative common difference must not be ignored: decreasing by 10 means adding \(-10\). Using \(a_n=a_1+(n-1)d\), we get \(a_5=100+(5-1)(-10)=100-40=60\). The common mistake is to count the starting term as a gap; there are four moves from the first term to the fifth term. Memory cue: the fifth term is reached after four differences, not five.

Correct answer: C. 4

Explanation: The governing concept is that the common difference of an arithmetic progression is found by subtracting any term from the next consecutive term. Using the first two terms, d = (x + 4) − x. On simplifying, the x and −x cancel, leaving d = 4. The next pair confirms the same result: (x + 8) − (x + 4) = x + 8 − x − 4 = 4. Therefore option C is correct. Option A incorrectly treats the variable part as the difference. Option D is the difference between the first and third terms, not consecutive terms, because (x + 8) − x = 8. Option B is not obtained from any valid consecutive subtraction. The common difference is therefore constant and independent of x.

Correct answer: A. (0)

Explanation: Consecutive terms are equal, so the difference is (3-3=0). In exams, a constant sequence can also be an arithmetic progression.

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

Explanation: An arithmetic progression is formed by repeatedly adding the same common difference. The first term is \(a=9\), and the common difference is \(d=0\). Therefore the second term is \(9+0=9\), and the third term is again \(9+0=9\). In fact, adding zero never changes a number, so every term of this progression remains 9.

The first option alternates incorrectly, while the second and fourth options show sequences with nonzero changes. The defining feature here is not merely the starting value 9 but the instruction that the difference is zero. Hence the first three terms are \(9,9,9\), which is exactly option C. The supplied answer and the calculation agree.

Correct answer: C. (6)

Explanation: The difference between (10) and (14) is (4), so the missing term is (10-4=6). In exams, apply the common difference backward too.

Correct answer: C. (5)

Explanation: In the linear rule (a_n=5n-2), the coefficient of (n) is the difference (5). In exams, take (k) as (d) in (kn+c).

Correct answer: B. 27

Explanation: Here, the first term is \(a=12\) and the common difference is \(d=15-12=3\). The \(n\)th term of an arithmetic progression is \(a_n=a+(n-1)d\). Therefore, \(a_6=12+(6-1)\times3=27\). Option 30 results from incorrectly using \(n\) instead of \(n-1\). Exam tip: Since the first term is \(a_1\), the formula contains \(n-1\).

Correct answer: A. 2, 5, 8, 11, \(\ldots\)

Explanation: In option A, consecutive differences are \(5-2=3\), \(8-5=3\), and \(11-8=3\), so it is an AP. In option D, the differences change. Exam tip: always check consecutive differences.

Correct answer: B. (10, 8, 6, 4)

Explanation: In the sequence (10, 8, 6, 4), 2 is subtracted to get each next term: 8 - 10 = -2, 6 - 8 = -2, and 4 - 6 = -2. Hence, its common difference is -2. Options A and C have positive common differences, while option D has common difference 0. Exam tip: Find the common difference by calculating next term − previous term.

Correct answer: B. 5

Explanation: Given \(a_n=11-2n\). Substituting \(n=3\), we get \(a_3=11-2(3)=11-6=5\). Therefore, option B is correct. The value \(7\) would result from subtracting only \(n\) instead of \(2n\), which does not follow the given rule. Exam tip: Substitute the value of \(n\) first, then perform multiplication and subtraction.

Correct answer: B. (a_n=11-2n)

Explanation: At (n=1) it gives (9), and at (n=2) it gives (7), so (a_n=11-2n). In exams, choose the rule after checking the first term.

Correct answer: C. (10)

Explanation: The middle term is the average of (5) and (15), so (x=\frac{5+15}{2}=10). In exams, take the average for the middle term in three-term progressions.

Correct answer: D. 54

Explanation: In this arithmetic progression, the first term is \(a=18\) and the common difference is \(d=24-18=6\). The \(n\)th term is \(a_n=a+(n-1)d\). Therefore, \(a_7=18+(7-1)\times6=18+36=54\). Hence, 54 is correct. The value 48 is the sixth term because the common difference is added only five times. Exam tip: for the \(n\)th term, use \((n-1)\), not \(n\).

Correct answer: A. (6,10,14,18,\ldots)

Explanation: An arithmetic progression is a sequence in which the same number is added to obtain each next term. The first term is the term at the beginning, and the common difference is found by subtracting one term from the next. In option A, the sequence starts with 6 and continues as 10, 14, and 18. Each step adds 4, so it has first term 6 and common difference 4.

The other choices fail at least one condition. Option B starts with 4, not 6, although its difference is 4. Option C starts with 6, but its difference is 6 because 12-6=6. Option D has a difference of 4 but starts with 10. Thus only option A satisfies both requirements simultaneously. A quick check is \(10-6=4\), \(14-10=4\), and \(18-14=4\).

Correct answer: A. 4, 4, 4, 4, …

Explanation: In an arithmetic progression, d is the common difference between consecutive terms. For option A, every term is 4, so 4 − 4 = 0 throughout and d = 0. Option B has d = 1, option C has d = 5, and option D has d = −3. A constant sequence is therefore a valid arithmetic progression with zero common difference, making option A correct.

Correct answer: B. 28

Explanation: This is an AP with first term 18 and common difference 2. The 6th term is 18 + (6 - 1) × 2 = 28. Choosing 30 gives the 7th-row value instead. Exam tip: always use n - 1 in the nth-term formula.