Arithmetic Progression

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

The governing idea is that an explicit rule gives each term directly. Substitute n = 1, 2, 3, 4, and 5 into aₙ = 4n + 7: the terms are 11, 15, 19, 23, and 27. Their sum is 11 + 15 + 19 + 23 + 27 = 95. Equivalently, this is an arithmetic progression with first term 11 and common difference 4, so S₅ = 5/2[2(11) + 4(5 − 1)] = 5/2(22 + 16) = 95. Thus option C is correct. The other choices arise from omitting a term, making an addition error, or using an incorrect first term or difference.

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 \(d\).

(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.

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.

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.

(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.

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.

(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.

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

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.

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 \(d\).

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.

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.

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 \(n\).

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.

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.

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).

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.

(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.