Arithmetic Progression

Correct answer: B. 4

Explanation: In an arithmetic progression, the difference between two terms equals the difference in their positions multiplied by the common difference. Thus, \(a_{12}-a_5=(12-5)d\), so \(46-18=7d\). Hence, \(28=7d\) and \(d=4\). Option 7 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 initial term.

Correct answer: A. 1

Explanation: In an arithmetic progression, the third term is \(a_3=a_1+2d\). Hence, \(11=a_1+2(5)=a_1+10\), so \(a_1=1\). If 6 were the first term, the third term would be \(6+10=16\), not 11. Exam tip: use \(a_n=a_1+(n-1)d\) to relate any term to the first term.

Correct answer: B. (2)

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

Correct answer: C. 24

Explanation: In an arithmetic progression, the difference between two terms equals the difference in their positions multiplied by the common difference. Thus, \(a_9-a_2=7d=44-9=35\), so \(d=5\). Now, \(a_5=a_2+3d=9+3\times5=24\). Option 22 does not fit a progression with common difference 5. Exam tip: first find \(d\) from the given terms, then move to the required term.

Correct answer: A. \(2y=x+z\)

Explanation: Consecutive terms have equal differences, so \(y-x=z-y\). Rearranging gives \(2y=x+z\); thus, the middle term is the arithmetic mean of its neighbours. \(y=x+z\) is not generally true. In exams, use this relation to identify an AP quickly.

Correct answer: B. (5)th

Explanation: The terms are (18,13,8,3,-2,\ldots), so the first negative term is the (5)th. Check terms around zero in order.

Correct answer: B. (56)

Explanation: There are (8) gaps between (a_{12}) and (a_4), and (d=7), so the difference is (56). Term difference depends on position difference.

Correct answer: C. (111)

Explanation: The first (6) terms are (11,14,17,20,23,26), and their sum is (111). Forming terms from the rule and adding is a safe method.

Correct answer: C. 6

Explanation: For an arithmetic progression, \(a_n=a_1+(n-1)d\). Thus, \(a_3=a_1+2d=16\) and \(a_8=a_1+7d=41\). Subtracting the equations gives \(5d=25\), so \(d=5\). Now, \(a_1=16-2\times5=6\). Therefore, 6 is correct. Option 5 is a close distractor because it is the common difference, not the first term. Exam tip: When two terms are given, subtract their equations first to find \(d\).

Correct answer: B. (25)

Explanation: In an arithmetic progression, the average of (a_4) and (a_8) is (a_6), so (a_6=\frac{50}{2}=25). Equidistant terms average to the middle term.

Correct answer: C. (6)

Explanation: From (2(x+2)=(x-4)+(2x-2)), (x=8), so the common difference is (10-4=6). First find (x), then find the difference.

Correct answer: D. (32)

Explanation: (a_6) is equidistant from (a_2) and (a_{10}), so (a_6=\frac{64}{2}=32). The average of symmetric terms is the middle term.

Correct answer: B. (5)

Explanation: Using (a_4=3+3d) and (a_6=3+5d), (6+8d=56), so (d=\frac{25}{4}). This result is not among the options.

Correct answer: C. (97)

Explanation: The general term is (a_n=3n+4), and the greatest value below (100) is (97). In boundary questions check the nearest terms.

Correct answer: C. 30

Explanation: Given \(a_n=40-5n\), we get \(a_3=40-5(3)=25\) and \(a_7=40-5(7)=5\). Therefore, \(a_3+a_7=25+5=30\). The value 25 is only \(a_3\), not the required sum. Exam tip: Substitute each required value of \(n\) separately before adding the terms.

Correct answer: A. -5

Explanation: Use the difference between the two given terms:

a_{11}-a_6=(a_1+10d)-(a_1+5d)=5d.

Thus,

5d=55-25=30, so d=6. Now, from

a_6=a_1+5d, we get a_1=25-5(6)=-5. Hence, option A is correct. If a_1 were -4, the sixth term would be 26, not 25. Exam tip: when two AP terms are given, first use the difference in their term numbers to find d.

Correct answer: B. 10

Explanation: For three consecutive terms of an arithmetic progression, twice the middle term equals the sum of the first and third terms. Thus, \(2(a+8)=a+(3a-4)\). This gives \(2a+16=4a-4\), so \(2a=20\) and hence \(a=10\). Therefore, option B is correct. If \(a=12\), the differences between the three terms are not equal. Exam tip: for three consecutive AP terms, use \(2\times\text{middle term}=\text{first term}+\text{third term}\).

Correct answer: C. Sixth term

Explanation: To find the zero term, set the given term equal to 0: \(12-2n=0\). Thus, \(2n=12\), so \(n=6\). Hence, the sixth term of the AP is zero. At the fifth term, \(12-2(5)=2\), 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. (22)

Explanation: The increase over six gaps is (24), so (d=4) and (a_7=10+3(4)=22). (a_7) lies between the two given terms.

Correct answer: A. (16,12,8,4,\ldots)

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

Correct answer: B. (54)

Explanation: There are (6) gaps between (a_{15}) and (a_9), so the difference is (6\times9=54). Term differences can be found directly using (d).

Correct answer: C. (107)

Explanation: The general term is (a_n=7n+2), and (a_{15}=107) is the first term greater than (100). In boundary questions check nearby terms.

Correct answer: C. (36)

Explanation: (a_6) is equidistant from (a_3) and (a_9), so (a_6=\frac{72}{2}=36). The average of symmetric terms is the middle term.

Correct answer: C. 76

Explanation: Here, the first term is \(a=3\) and the common difference is \(d=7\). Using \(a_n=a+(n-1)d\), we get \(a_4=3+3\times7=24\) and \(a_8=3+7\times7=52\). Therefore, \(a_4+a_8=24+52=76\). The value 74 can result from not applying the common difference of 7 correctly. Exam tip: when finding the \(n\)th term of an AP, use \((n-1)d\), not \(nd\).

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, \(a_{13}-a_5=(13-5)d=8d\). Hence, \(65-17=48=8d\), giving \(d=6\). Option 8 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 initial term.