Arithmetic Progression

Correct answer: B. (4)

Explanation: The first (6) terms sum to (72) for (2,6,10,14,18,22), so (d=4). For small (n), option checking is quick.

Correct answer: C. 13

Explanation: Here, the first term is 12 and the common difference is 5. Therefore, the nth term is \(a_n=12+(n-1)\times5=5n+7\). Substituting \(a_n=72\), we get \(5n+7=72\), so \(5n=65\) and \(n=13\). Thus, 72 is the 13th term of the progression. The nearby option 14 is incorrect because the 14th term is \(77\). Exam tip: To find a term position, first use \(a_n=a+(n-1)d\).

Correct answer: C. (36)

Explanation: The increase over three gaps is (12), so (d=4) and (a_9=20+4(4)=36). Moving forward from a nearby term is easy.

Correct answer: C. (46)

Explanation: The general term is (a_n=4n+2), and the greatest term below (50) is (46). In boundary questions check nearby terms.

Correct answer: A. 16

Explanation: In an arithmetic progression, \(a_n=a_1+(n-1)d\). Thus, \(a_2=20+(2-1)(-4)=16\) and \(a_6=20+(6-1)(-4)=0\). Therefore, \(a_2+a_6=16+0=16\), so option A is correct. The value 20 may result from not applying the common difference \(-4\) correctly to the terms. Exam tip: Write \((n-1)d\) before substituting values for any term.

Correct answer: A. -5

Explanation: In an arithmetic progression, the difference between two terms equals the difference of their positions multiplied by the common difference. Thus, \(a_{12}-a_3=9d=61-7=54\), so \(d=6\). Now, from \(a_3=a_1+2d\), we get \(7=a_1+12\), hence \(a_1=-5\). If \(-4\) were used, the third term would be \(8\), not \(7\). Exam tip: first find \(d\) from the two given terms, then use \(a_n=a_1+(n-1)d\).

Correct answer: B. (189)

Explanation: The first and ninth terms are (1) and (41), so the sum is (\frac{9(1+41)}{2}=189). Use the first and last terms for the sum.

Correct answer: C. (9)th

Explanation: The answer is option C, the 9th term. We need the first term for which \(a_n<0\). Given \(a_n=25-3n\), test the boundary: \(25-3n<0\), so \(25<3n\), hence \(n>25/3=8\frac{1}{3}\). The smallest whole-number position is \(n=9\). Direct checking gives \(a_8=25-24=1\), still positive, and \(a_9=25-27=-2\), negative. Option A, 7th, gives 4, positive. Option B, 8th, gives 1, positive. Option C, 9th, gives -2 and is the first negative term. Option D, 10th, gives -5, which is negative but not the first one. The key is to use a strict inequality because “negative” means less than zero, not equal to zero. Memory cue: find the first integer after the expression crosses below zero.

Correct answer: C. (20)

Explanation: (a_4) is equidistant from (a_2) and (a_6), so (a_4=\frac{40}{2}=20). The average of symmetric terms is the middle term.

Correct answer: C. 58

Explanation: The nth term of an arithmetic progression is \(a_n=a_1+(n-1)d\). Thus, \(a_3=9+(3-1)\times4=17\) and \(a_9=9+(9-1)\times4=41\). Therefore, \(a_3+a_9=17+41=58\), so option C is correct. Choosing 56 may result from an error in finding the term number or multiplying the common difference. Exam tip: always add \((n-1)d\) to the first term when finding the nth term.

Correct answer: D. 86

Explanation: In this arithmetic progression, the first term is 8 and the common difference is 7. Thus, \(a_{10}=8+(10-1)\times7=71\) and \(a_2=8+(2-1)\times7=15\). Hence, \(a_{10}+a_2=71+15=86\). A value such as 84 may result from using an incorrect common difference or term position. Exam tip: always use \(n-1\) in \(a_n=a+(n-1)d\).

Correct answer: C. (66)

Explanation: The increase over five gaps is (30), so (d=6) and (a_{12}=48+3(6)=66). First find (d), then move forward.

Correct answer: B. \(5\)

Explanation: For an arithmetic progression, \(a_{14}-a_6=(14-6)d\). Thus, \(61-21=8d\), so \(40=8d\) and hence \(d=5\). If \(d=4\), the difference across 8 term positions would be \(32\), not \(40\). Exam tip: When subtracting two AP terms, also subtract their term numbers.

Correct answer: C. 11th term

Explanation: The nth term is \(a_n=8n-5\). To find the position of 83, set \(8n-5=83\). This gives \(8n=88\), so \(n=11\). Hence, 83 is the 11th term. The 10th term is \(75\), so it is a close but incorrect option. Exam tip: to find the position of a given term, equate \(a_n\) to that value and solve for \(n\).

Correct answer: C. (6)

Explanation: For three consecutive terms, (2(2x+1)=(x-3)+(4x-1)), giving (x=6). The middle term is the average of the two surrounding terms.

Correct answer: A. The difference between any two consecutive terms is the same

Explanation: In an AP, a fixed common difference d is added to each term, so aₙ₊₁−aₙ=d. A constant ratio identifies a geometric progression, not necessarily an AP. Exam tip: compare consecutive differences to identify an AP.

Correct answer: B. (5,11,17,23,\ldots)

Explanation: In (5,11,17,23,29,\ldots), the fifth term is (29) and the common difference is (6). Check each option up to the required term.

Correct answer: B. 6

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Thus, \(73=7+(12-1)d=7+11d\). Hence \(11d=66\), so \(d=6\). There are 11 gaps from the 1st term to the 12th term, not 12. Exam tip: always use \(n-1\) in the formula for the \(n\)th term.

Correct answer: B. (6)th

Explanation: The terms are (25,19,13,7,1,-5,\ldots), so the first negative term is the (6)th. Check terms around zero in order.

Correct answer: B. (63)

Explanation: There are (9) gaps between (a_{15}) and (a_6), and (d=7), so the difference is (63). Term difference depends on position difference.

Correct answer: B. 4

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Thus, \(a_4=a_1+3d=22\) and \(a_9=a_1+8d=52\). Subtracting the equations gives \(5d=30\), so \(d=6\). Now, \(a_1=22-3\times6=4\). Therefore, the correct answer is 4. Option 6 is the common difference \(d\), not the first term. Exam tip: When two terms are given, subtract their equations first to find \(d\).

Correct answer: C. (43)

Explanation: (a_9) is equidistant from (a_5) and (a_{13}), so (a_9=\frac{86}{2}=43). The average of symmetric terms is the middle term.

Correct answer: D. (8)

Explanation: From (2(3x+2)=(2x-1)+(5x+1)), (x=4), so the common difference is (14-7=7). Calculate (x) first and then the difference.

Correct answer: B. (5)

Explanation: The terms are (18,14,10,6,2,-2,\ldots), so there are (5) positive terms. Do not count zero or negative terms as positive.

Correct answer: C. (54)

Explanation: (a_9) is equidistant from (a_3) and (a_{15}), so (a_9=\frac{108}{2}=54). The average of symmetric terms is the middle term.