Arithmetic Progression

Correct answer: B. (2,9,16,23,\ldots)

Explanation: In (2,9,16,23,\ldots), (a_4) is not (19); the correct sequence would be (-2,5,12,19,\ldots). None of the given options is correct.

Correct answer: A. (6)

Explanation: (a_4=6+3d) and (a_8=6+7d), so (12+10d=84), giving (d=7.2), which is not in the options. Option consistency must be checked.

Correct answer: B. (86)

Explanation: The general term is (a_n=5n+6), and the greatest term below (90) is (86). In boundary questions, check nearby terms.

Correct answer: B. 52

Explanation: Given \(a_n=52-4n\), \(a_4=52-4(4)=36\) and \(a_9=52-4(9)=16\). Therefore, \(a_4+a_9=36+16=52\). The option 48 may result from substituting an incorrect value of \(n\) in a term. Exam tip: substitute the term number carefully into the formula before simplifying.

Correct answer: A. -8

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Thus, \(a_{12}-a_7=5d=69-34=35\), so \(d=7\). Now, using \(a_7=a_1+6d\), we get \(34=a_1+42\), hence \(a_1=-8\). If \(-6\) were chosen, the seventh term would be \(36\), not the given \(34\). Exam tip: When two terms are given, first use the difference in their term numbers to find \(d\).

Correct answer: C. (11)

Explanation: From (2(p+10)=p+(3p-2)), (2p+20=4p-2), so (p=11). The middle term is the average of the two surrounding terms.

Correct answer: C. (256)

Explanation: The first (8) terms go from (4) to (60), so the sum is (\frac{8(4+60)}{2}=256). Pairing symmetric terms makes calculation quick.

Correct answer: C. Sixth term

Explanation: To find the zero term, set the given term equal to 0: \(18-3n=0\). Thus, \(3n=18\), so \(n=6\). Therefore, the sixth term is zero. For example, the fifth term is \(a_5=18-15=3\), so it is not zero. Exam tip: when a term with a specified value is asked, equate \(a_n\) to that value and solve for \(n\).

Correct answer: C. (32)

Explanation: The increase over six gaps is (36), so (d=6) and (a_8=14+3(6)=32). (a_8) lies between the two given terms.

Correct answer: A. (24,20,16,12,\ldots)

Explanation: In (24,20,16,12,8,4,0,\ldots), the first term is (24) and the seventh term is (0). Check options up to the seventh term.

Correct answer: B. (72)

Explanation: There are (9) gaps between (a_{14}) and (a_5), so the difference is (9\times8=72). Term differences can be found directly using (d).

Correct answer: B. (104)

Explanation: The general term is (a_n=7n+6), and (a_{14}=104) is the first term greater than (100). In boundary questions, check nearby terms.

Correct answer: C. (56)

Explanation: (a_8) is equidistant from (a_4) and (a_{12}), so (a_8=\frac{112}{2}=56). The average of symmetric terms is the middle term.

Correct answer: B. 94

Explanation: The first term is \(a=5\) and the common difference is \(d=12-5=7\). Using \(a_n=a+(n-1)d\), we get \(a_5=5+4\times7=33\) and \(a_9=5+8\times7=61\). Therefore, \(a_5+a_9=33+61=94\). A nearby option such as 98 may result from using an incorrect term number or common difference. Exam tip: first identify \(a\) and \(d\), then apply the formula for \(a_n\).

Correct answer: B. 6

Explanation: In an arithmetic progression, the difference between two terms equals the difference of their indices multiplied by the common difference. Thus, \(a_{15}-a_6=(15-6)d=9d\). Hence, \(82-28=54=9d\), giving \(d=6\). Option 9 is the difference between the term numbers, not the common difference. Exam tip: use \(a_m-a_n=(m-n)d\) to find \(d\) without first finding the first term.

Correct answer: B. 3

Explanation: For three consecutive terms of an arithmetic progression, twice the middle term equals the sum of the first and third terms. Thus, \(2(5x-2)=(3x+1)+(8x-8)\). This gives \(10x-4=11x-7\), so \(x=3\). On checking, the terms are \(10,13,16\), with common difference \(3\). Exam tip: for three consecutive AP terms, use \(2b=a+c\) directly.

Correct answer: C. (6)

Explanation: Direct answer: Option C, 6. For an arithmetic progression, the sum of the first n terms is \\(S_n=\\frac n2[2a+(n-1)d]\\). Here \\(a=5\\), \\(n=6\\), and \\(S_6=120\\). Substitute carefully: \\(120=\\frac62[2(5)+(6-1)d]\\). Thus \\(120=3(10+5d)=30+15d\\). Subtract 30: \\(90=15d\\). Divide by 15: \\(d=6\\). Option A, 4, would give sum \\(3(10+20)=90\\), not 120. Option B, 5, would give \\(3(10+25)=105\\). Option C, 6, gives \\(3(10+30)=120\\), so it is correct. Option D, 7, would give \\(3(10+35)=135\\). A direct check gives terms 5, 11, 17, 23, 29, 35, whose sum is 120. Remember: in the sum formula use n−1, not n, and solve the equation step by step.

Correct answer: C. उभयनिष्ठ अंतर

Explanation: In an arithmetic progression, subtracting a term from the next term gives one fixed value, called the common difference. The first term is only the starting term. Exam tip: subtract consecutive terms to identify an AP.

Correct answer: C. (60)

Explanation: The increase over four gaps is (24), so (d=6) and (a_{11}=36+4(6)=60). Moving forward from a nearby term is easy.

Correct answer: C. (74)

Explanation: The general term is (a_n=6n+2), and the greatest term below (80) is (74). In boundary questions, check nearby terms.

Correct answer: B. 20

Explanation: The nth term of an arithmetic progression is \(a_n=a_1+(n-1)d\). Thus, \(a_3=30+2(-5)=20\) and \(a_7=30+6(-5)=0\). Therefore, \(a_3+a_7=20+0=20\). Choosing 25 can result from incorrectly calculating the number of common differences. Exam tip: always use \((n-1)d\) while finding \(a_n\).

Correct answer: A. -5

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Thus, \(a_{13}-a_4=9d=67-13=54\), so \(d=6\). Now, using \(a_4=a_1+3d\), we get \(13=a_1+18\), hence \(a_1=-5\). If \(-4\) were the first term, the fourth term would be \(14\), so it is not correct. Exam tip: subtracting two given terms is a quick way to find the common difference \(d\).

Correct answer: C. (300)

Explanation: The first and tenth terms are (3) and (57), so the sum is (\frac{10(3+57)}{2}=300). Use the first and last terms for the sum.

Correct answer: C. (8)th

Explanation: The direct answer is C: the 8th term. A sequence is negative when its value is less than zero. Here the nth term is given by \\(a_n=31-4n\\). Test the terms in order: \\(a_6=31-4(6)=31-24=7\\), so the 6th term is positive; \\(a_7=31-28=3\\), so the 7th term is still positive; \\(a_8=31-32=-1\\), so the 8th term is negative. Thus the first negative term is the 8th term. A is wrong because the 6th term is 7. B is wrong because the 7th term is 3. C is correct because its value is -1. D is not the first negative term; although \\(a_9=-5\\) is also negative, negativity already began at term 8. Exam cue: calculate consecutive terms until the first value below zero.

Correct answer: C. (30)

Explanation: (a_5) is equidistant from (a_3) and (a_7), so (a_5=\frac{60}{2}=30). The average of symmetric terms is the middle term.