Arithmetic Progression

Correct answer: B. 82

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Thus, \(a_4=11+3\times5=26\) and \(a_{10}=11+9\times5=56\). Therefore, \(a_4+a_{10}=26+56=82\), so option B is correct. A value such as 84 can result from using one extra multiple of the common difference while finding a term. Exam tip: always use \((n-1)d\) for the \(n\)th term.

Correct answer: B. 108

Explanation: For this AP, the first term is \(a=12\) and the common difference is \(d=19-12=7\). Using \(a_n=a+(n-1)d\), \(a_{11}=12+10\times7=82\) and \(a_3=12+2\times7=26\). Therefore, \(a_{11}+a_3=82+26=108\). The nearby value 112 can result from an error in the term number or common difference. Exam tip: always use \(n-1\) in the formula for \(a_n\).

Correct answer: C. (80)

Explanation: The increase over five gaps is (35), so (d=7) and (a_{13}=59+3(7)=80). First find (d), then move forward.

Correct answer: B. (15)

Explanation: (a_2=a_1+5) and (a_8=a_1+35), so (2a_1+40=70) and (a_1=15). In sum-based questions, write both terms using (a_1) and (d).

Correct answer: C. 6

Explanation: In an arithmetic progression, the difference between two terms equals the difference in their positions multiplied by the common difference. Thus, \(a_{13}-a_4=(13-4)d=9d\). Here, \(73-19=54\), so \(9d=54\) and \(d=6\). Note that 9 is the difference between the term numbers, not the common difference. Exam tip: Use \(a_n-a_m=(n-m)d\) to find \(d\) without first finding the first term.

Correct answer: C. 13th term

Explanation: Given \(a_n=9n-4\). To find the position of 113, put \(a_n=113\): \(9n-4=113\Rightarrow 9n=117\Rightarrow n=13\). Therefore, 113 is the 13th term of the progression. The 12th term is \(9(12)-4=104\), so it is not correct. Exam tip: To find the position of a given term, equate \(a_n\) to that term's value.

Correct answer: B. (6)

Explanation: For three consecutive terms, (2(3x+6)=(2x+1)+(5x+5)), giving (x=6). The middle term is the average of the two surrounding terms.

Correct answer: C. 44

Explanation: In an arithmetic progression, the common difference is constant. Here, \(a_{11}-a_3=8d=72-16=56\), so \(d=7\). Therefore, \(a_7=a_3+4d=16+4\times7=44\). Option 42 is incorrect because it does not give the required difference across four terms from \(a_3\) to \(a_7\). Exam tip: When a term’s index is midway between two given indices, its value is also the average of those two terms.

Correct answer: C. (13,18,23,28,\ldots)

Explanation: In (13,18,23,28,33,38,\ldots), the sixth term is (38) and the common difference is (5). Check each option up to the required term.

Correct answer: B. 6

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Hence, \(93=9+(15-1)d=9+14d\). Therefore, \(14d=84\), so \(d=6\). If 7 were used, the 15th term would be \(9+14\times7=107\), not 93. Exam tip: there are always \(n-1\) gaps before the \(n\)th term.

Correct answer: C. (8)th

Explanation: The terms are (31,25,19,13,7,1,-5,\ldots), so the first negative term is the (7)th. Among the options, the (7)th is correct.

Correct answer: B. (72)

Explanation: There are (9) gaps between (a_{14}) and (a_5), and (d=8), so the difference is (72). Term difference depends on position difference.

Correct answer: C. (159)

Explanation: The first (6) terms are (14,19,24,29,34,39), and their sum is (159). Forming terms from the rule and adding is a safe method.

Correct answer: B. 2

Explanation: In an arithmetic progression, the difference between two terms equals the difference in their positions multiplied by the common difference. Thus, \(a_{12}-a_5=7d=68-26=42\), so \(d=6\). Now \(a_5=a_1+4d\), hence \(26=a_1+4(6)\), giving \(a_1=2\). Option 6 is the common difference, not the first term. Exam tip: use \(a_n=a_1+(n-1)d\), taking care to use \(n-1\).

Correct answer: C. (60)

Explanation: (a_{10}) is equidistant from (a_6) and (a_{14}), so (a_{10}=\frac{120}{2}=60). The average of symmetric terms is the middle term.

Correct answer: D. (12)

Explanation: From (2(4x+3)=(3x-2)+(6x+1)), (x=7), so the terms are (19,31,43) and the difference is (12). First find (x), then find the difference.

Correct answer: B. (5)

Explanation: The terms are (22,17,12,7,2,-3,\ldots), so there are (5) positive terms. Do not count zero or negative terms as positive.

Correct answer: C. (70)

Explanation: (a_{10}) is equidistant from (a_4) and (a_{16}), so (a_{10}=\frac{140}{2}=70). The average of symmetric terms is the middle term.

Correct answer: A. (1,9,17,25,\ldots)

Explanation: The direct answer is A: \((1,9,17,25,\ldots)\). For an arithmetic progression, \(a_n=a+(n-1)d\). The given terms differ by \(65-17=48\) across six index steps, from the third term to the ninth term. Hence \(6d=48\), so \(d=8\). Since \(a_3=a+2d=17\), we get \(a+16=17\), so \(a=1\). The progression is therefore \(1,9,17,25,33,41,49,57,65\). Option A has first term 1 and common difference 8, giving the required third and ninth terms. Option B has difference 7, so its ninth term is 59, not 65. Option C has difference 6, so its ninth term is 53. Option D has first term 9 and difference 8; its third term is 25, not 17. Always test both given conditions, not just one. Memory cue: the gap between the third and ninth terms contains six equal AP gaps.

Correct answer: B. 6

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Hence, \(a_5=4+4d\) and \(a_9=4+8d\). Using the given condition, \((4+4d)+(4+8d)=80\), so \(8+12d=80\). Therefore, \(12d=72\) and \(d=6\). If the common difference were 5, the sum would be \(68\), not 80. Exam tip: first express each given term in terms of \(a_1\) and \(d\).

Correct answer: C. (119)

Explanation: The general term is (a_n=5n+9), and the greatest term below (120) is (119). In boundary questions, check nearby terms.

Correct answer: B. 61

Explanation: Given \(a_n=63-5n\), we get \(a_4=63-5(4)=43\) and \(a_9=63-5(9)=18\). Therefore, \(a_4+a_9=43+18=61\), so option B is correct. A value such as 66 may result from substituting the term numbers incorrectly. Exam tip: always substitute the value of \(n\) in brackets in the nth-term formula.

Correct answer: A. -4

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Thus, \(a_{13}-a_8=5d=80-45=35\), so \(d=7\). Now, from \(a_8=a_1+7d\), we get \(45=a_1+49\), hence \(a_1=-4\). If \(-2\) were the first term, the eighth term would be \(47\), not \(45\). Exam tip: subtract the given terms first to find \(d\).

Correct answer: C. 7

Explanation: For three consecutive terms of an arithmetic progression, twice the middle term equals the sum of the first and third terms. Thus, \(2(2p+5)=(p+2)+(4p+1)\). This gives \(4p+10=5p+3\), so \(p=7\). If 5 is substituted, the three terms do not have a common difference. Exam tip: for three consecutive AP terms, use \(2b=a+c\) directly.

Correct answer: C. (333)

Explanation: The first and ninth terms are (5) and (69), so the sum is (\frac{9(5+69)}{2}=333). Pairing symmetric terms makes calculation quick.