Arithmetic Progression

Correct answer: D. (33)

Explanation: From (a_4) to (a_9), there are (5) gaps, so (a_9=18+5(3)=33). Moving forward from a nearby term is a quick method.

Correct answer: B. 11

Explanation: To find the first term, substitute \(n=1\). Thus, \(a_1=9(1)+2=11\), so 11 is correct. Option 9 is only the coefficient of \(9n\); the constant term 2 must also be added. Exam tip: To obtain the first term from any \(a_n\), always put \(n=1\).

Correct answer: A. (5,7,9,11)

Explanation: The direct answer is option A: \((5,7,9,11)\). In an arithmetic progression, the first term is \(a=5\), and the common difference is \(d=2\), meaning add 2 to obtain each next term. Thus the first term is 5, the second is \(5+2=7\), the third is \(7+2=9\), and the fourth is \(9+2=11\). Option A follows both given conditions exactly. Option B starts with 2, not 5, and its difference is 3. Option C starts with 5 but adds 5 each time, so its common difference is 5 rather than 2. Option D starts with 7, not the required first term, although its difference is 2. The formula \(a_n=a+(n-1)d\) also gives \(a_n=5+2(n-1)\), confirming the list. Memory cue: first write the given first term, then repeatedly add the common difference.

Correct answer: C. (4)

Explanation: From the third to the fifth term there are (2) gaps and the increase is (8), so (d=4). Count the number of gaps between terms.

Correct answer: A. (a_n=26-4n)

Explanation: The direct answer is option A: \(a_n=26-4n\). The first term is 22, and the common difference is \(18-22=-4\). The standard AP formula is \(a_n=a_1+(n-1)d\). Substitution gives \(a_n=22+(n-1)(-4)=22-4n+4=26-4n\). Checking: at \(n=1\), the formula gives 22; at \(n=2\), it gives 18; at \(n=3\), it gives 14; and at \(n=4\), it gives 10. Option A therefore matches. Option B, \(22-4n\), gives 18 at \(n=1\), so it starts incorrectly. Option C, \(4n+18\), increases and gives 22, 26, 30, so it has the wrong direction. Option D, \(26+4n\), also increases and gives 30 first. The important point is that a decreasing AP has a negative common difference. Memory cue: simplify \(a_1+(n-1)d\) carefully; the extra \(+4\) changes 22 into 26.

Correct answer: C. (8)

Explanation: The average of the first three terms is (\frac{2+8+14}{3}=8). In an arithmetic progression, the average of three consecutive terms is the middle term.

Correct answer: C. 335

Explanation: This is an AP with \(a=20\), \(d=3\), and \(n=10\). So, \(S_{10}=\frac{10}{2}[2(20)+9(3)]=5(67)=335\). Finding only the 10th term does not give the total seats. Exam tip: use \(S_n\) when a total is asked.

Correct answer: C. 47

Explanation: The first term is \(a=17\), and the common difference is \(d=23-17=6\). The \(n\)th term is \(a_n=a+(n-1)d\). Thus, \(a_6=17+(6-1)\times6=17+30=47\). Therefore, 47 is correct. Note that 41 is the fifth term, not the sixth. Exam tip: add the common difference \(n-1\) times to obtain the \(n\)th term.

Correct answer: D. 36

Explanation: Given \(a_n=5n+3\), \(a_2=5(2)+3=13\) and \(a_4=5(4)+3=23\). Therefore, \(a_2+a_4=13+23=36\). The option 34 can result from an error while evaluating one of the terms. In exams, find each required term first and then add them.

Correct answer: C. 11th term

Explanation: Here, the first term is \(a=50\) and the common difference is \(d=-5\). The \(n\)th term is \(a_n=a+(n-1)d\). Thus, \(0=50+(n-1)(-5)\), giving \(n-1=10\) and hence \(n=11\). The 10th term is \(5\), not zero. Exam tip: To find a term’s position, substitute the given term for \(a_n\) in \(a_n=a+(n-1)d\).

Correct answer: D. 65

Explanation: The formula for the nth term of an arithmetic progression is \(a_n=a+(n-1)d\). Therefore, \(a_8=2+(8-1)\times9=2+63=65\). Hence, 65 is correct. \(63\) is only the value of \(7d\); the first term \(a=2\) must also be added. Exam tip: for the nth term, the number of common differences is always \(n-1\).

Correct answer: C. (115)

Explanation: In an arithmetic progression, each term is found by adding the common difference to the previous term. The given progression starts at 3 and increases by 10, so its first five terms are 3, 13, 23, 33, and 43. To find their sum, add them directly: \(3+13+23+33+43=115\). Thus the correct choice is option C.

The same result follows from the arithmetic-series formula \(S_n=\frac{n}{2}(a+l)\), where n is the number of terms, a is the first term, and l is the last term. Here n=5, a=3, and l=43, so \(S_5=\frac{5}{2}(3+43)=\frac{5}{2}\times46=115\). The value 105 would omit some increase, while 110 and 125 do not equal the required sum.

Correct answer: A. 0

Explanation: For an arithmetic progression, the nth term is \(a_n=a_1+(n-1)d\). Thus, \(a_6=25+(6-1)(-5)=25-25=0\). Therefore, 0 is the correct answer. The option 5 may result from adding \(-5\) only four times instead of five times to reach the sixth term. Exam tip: always use \((n-1)\) common differences to find \(a_n\), not \(n\) differences.

Correct answer: B. (11,14,17,20,\ldots)

Explanation: Given \(a_n=3n+8\), putting \(n=1\) gives \(a_1=11\). Each successive term increases by 3, so the progression is \((11,14,17,20,\ldots)\). Option A also has common difference 3, but its first term is 8, so it is not correct. Exam tip: substitute \(n=1\) to find the first term, then check the common difference.

Correct answer: B. 8th

Explanation: Here, the first term is \(a=6\) and the common difference is \(d=15-6=9\). The \(n\)th term is \(a_n=a+(n-1)d\). So, \(69=6+(n-1)\times9\), giving \(n-1=7\) and \(n=8\). Hence, 69 is the 8th term. The 9th term would be \(78\), so option C is not correct. Exam tip: use \(n-1\), not \(n\), in the formula for the nth term of an AP.

Correct answer: B. (16)

Explanation: From (a_5) to (a_2), move back three gaps, so (a_2=28-3(4)=16). When moving backward, subtract the common difference.

Correct answer: D. (7)

Explanation: In an arithmetic progression, (2(x+3)=(x-2)+(2x-1)), which gives (x=9). The middle term is the average of the two adjacent terms.

Correct answer: B. (4)

Explanation: The direct answer is B: 4. In an arithmetic progression, the nth-term formula is a_n=a+(n-1)d . Here every displayed term is 4, so the common difference is d=4-4=0 and the first term is a=4 . For the twentieth term, a_{20}=4+(20-1)(0)=4+0=4 . Another simple way is to notice that adding zero changes nothing, so the sequence remains 4, 4, 4, 4 for every position. Option A, 0, is the common difference, not the twentieth term. Option B is correct because the twentieth entry is still 4. Option C, 20, confuses the term number with the term's value. Option D, 80, incorrectly multiplies 4 by 20; multiplication is not the rule for this constant progression. Exam cue: distinguish the position n from the value of the term, and calculate d before using the formula.

Correct answer: B. 18

Explanation: Given \(a_n=18-2n\), \(a_3=18-2(3)=12\) and \(a_6=18-2(6)=6\). Therefore, \(a_3+a_6=12+6=18\). A value such as 16 can result from substituting an incorrect value of \(n\) in one term. Exam tip: find each required term separately before adding them.

Correct answer: A. (a_n=7n+14)

Explanation: The direct answer is option A: \(a_n=7n+14\). The first term is 21 and the common difference is \(28-21=7\). Apply \(a_n=a_1+(n-1)d\): \(a_n=21+(n-1)7=21+7n-7=7n+14\). Checking gives 21 when \(n=1\), 28 when \(n=2\), 35 when \(n=3\), and 42 when \(n=4\). Thus option A is exact. Option B, \(14n+7\), gives 21 first but then 35, so its difference is 14 rather than 7. Option C, \(7n+21\), gives 28 at \(n=1\), so it starts one step late. Option D, \(21n+7\), has difference 21 and gives 28 first. Substituting \(n=1\) is a quick way to reject formulas, but also check the difference or a second term. Memory cue: first term plus \((n-1)\) times the common difference.

Correct answer: C. 37

Explanation: The nth term of an arithmetic progression is \(a_n=a+(n-1)d\). Here, \(a=9\), \(d=4\), and \(n=8\), so \(a_8=9+(8-1)\times4=9+28=37\). Therefore, 37 is the correct answer. An answer such as 35 usually results from using the wrong number of differences. Exam tip: multiply the common difference by \(n-1\), not by \(n\).

Correct answer: B. (a_n=39-4n)

Explanation: The first term is (35) and the difference is (-4), so (a_n=35+(n-1)(-4)=39-4n). In a decreasing progression the common difference is negative.

Correct answer: B. 34

Explanation: The formula for the nth term of an arithmetic progression is \(a_n=a+(n-1)d\). Thus, \(a_5=6+(5-1)\times7=6+28=34\). Hence, 34 is correct. The common difference is added only four times to reach the fifth term from the first term, so 35 does not follow the AP rule. Exam tip: for the nth term, use \(n-1\) common differences.

Correct answer: C. 9th term

Explanation: Here, the first term is \(a=8\) and the common difference is \(d=5\). The \(n\)th term is \(a_n=a+(n-1)d\). So, \(8+(n-1)\times5=48\) gives \(5(n-1)=40\), hence \(n=9\). Therefore, 48 is the 9th term. The 8th term is \(43\), so it is not correct. Exam tip: To find the position of a given term, equate it to \(a_n\) and solve for \(n\).

Correct answer: A. (5,8,11,14)

Explanation: Given \(a_n=3n+2\), substitute \(n=1,2,3,4\). This gives \(a_1=5\), \(a_2=8\), \(a_3=11\), and \(a_4=14\). Hence, the first four terms are \((5,8,11,14)\). Option C starts with 2, which is obtained by using \(n=0\), but the first term is found using \(n=1\). Exam tip: when a sequence is defined by \(a_n\), begin listing terms from \(a_1\).