Finding the sum of the first $n$ terms of an AP

Correct answer: C. (442)

Using (S_n=\frac{n}{2}[2a+(n-1)d]), we get (S_{13}=442). Write ((n-1)d) carefully in the formula.

Correct answer: C. (330)

The tenth term is (51), so (S_{10}=\frac{10}{2}(15+51)=330). Finding the last term first is an easy method.

Correct answer: B. (253)

The sum of the first (n) natural numbers is (\frac{n(n+1)}{2}), so the answer is (253). Put the value of (n) directly in such questions.

Correct answer: C. (225)

The sum of the first (n) odd numbers is (n^2), so (15^2=225). This short formula saves time in exams.

Correct answer: B. (380)

The sum of the first (n) even numbers is (n(n+1)), so (19\times20=380). Even numbers start from (2,4,6,\ldots).

Correct answer: C. 350

The governing concept is the sum of a finite arithmetic progression when the first term, last term, and number of terms are known. The efficient formula is S_n = n(a + l)/2. Here a = 8, l = 62, and n = 10. Substituting gives S_10 = 10(8 + 62)/2 = 10 × 70/2 = 350. Therefore option C is correct. The same reasoning follows from the fact that an AP is symmetric around the average of its first and last terms: the average is (8 + 62)/2 = 35, and ten terms therefore have sum 10 × 35 = 350. There is no need to find the common difference. The other options arise from using an incorrect average, term count, or arithmetic operation.

Correct answer: A. 208

The governing concept is the sum of the first n terms of an arithmetic progression. The first term is a = 40 and the common difference is d = 36 − 40 = −4, since the sequence decreases. The eighth term is a_8 = a + 7d = 40 + 7(−4) = 12. Using the endpoint form of the sum formula, S_8 = 8(a + l)/2 = 8(40 + 12)/2 = 4 × 52 = 208. Therefore option A is correct. The general formula confirms it: S_8 = 8/2[2(40) + 7(−4)] = 4(80 − 28) = 208. Retaining the negative sign in the common difference is essential; treating it as +4 produces an incorrect result and explains some distractors.

Correct answer: B. (243)

The ninth term is (47), so (S_9=\frac{9}{2}(7+47)=243). With an odd number of terms, the middle term can also check the sum.

Correct answer: B. (306)

Here (a=9), (d=3), and (n=12), so there will be (306) seats. In word problems, convert the pattern into an arithmetic progression.

Correct answer: C. (180)

The sum of the (7)th to (12)th terms is (S_{12}-S_6=180). Subtract partial sums to find the sum of middle terms.

Correct answer: A. (418)

The eleventh term is (73), so (S_{11}=\frac{11}{2}(3+73)=418). Finding the last term correctly is the key step.

Correct answer: B. (216)

The direct answer is option B, 216. Average means total sum divided by the number of terms: \(\text{average}=S/n\). Rearranging gives \(S=\text{average}\times n\). The average is 27 and there are 8 terms, so \(S_8=27\times8=216\). Option A, 206, is 10 too small. Option B, 216, is correct because it follows the definition of average exactly. Option C, 226, is also too large and has no valid calculation behind it. Option D, 236, is still larger and incorrect. No common difference or individual terms are needed here. The common mistake is to divide 27 by 8; instead, when the average and number of items are known, multiply them to obtain the sum.

Correct answer: B. (690)

The fifteenth term is (74), so (S_{15}=\frac{15}{2}(18+74)=690). The sum can also be found using the average of the first and last terms.

Correct answer: C. (220)

This is the sum of the first (10) multiples of (4), so the total is (220) questions. Treat the number of days as (n).

Correct answer: A. (77)

For an arithmetic progression, the first term is \\(a=20\\), the common difference is \\(d=-3\\), and the number of terms is \\(n=7\\). The seventh term is \\(a_7=a+(7-1)d=20+6(-3)=2\\). The sum formula is \\(S_n=\frac{n}{2}[2a+(n-1)d]\\), so \\(S_7=\frac{7}{2}[40+6(-3)]=\frac{7}{2}(22)=77\\).

The same result comes from pairing the first and last terms: the average is \\(\frac{20+2}{2}=11\\), and seven terms give \\(7\times11=77\\). Thus option A is correct. The negative difference decreases each term, but it must be included with its sign; replacing \\(d=-3\\) by 3 would produce an incorrect sum.

Correct answer: C. (630)

The twelfth term is (80), so (S_{12}=\frac{12}{2}(25+80)=630). Keep both the last term and (n) correct.

Correct answer: B. (169)

The sum of the first (n) odd numbers is (n^2), so (13^2=169). Remember this pattern for odd numbers.

Correct answer: B. 55

The governing concept is that a term formula must first be evaluated for each required index. For a_n = 3n + 2, substitute n = 1, 2, 3, 4, and 5 to obtain the terms 5, 8, 11, 14, and 17. These form an arithmetic progression with first term 5 and common difference 3. Their sum is 5 + 8 + 11 + 14 + 17 = 55. Alternatively, use S_5 = 5/2[2(5) + (5 − 1)(3)] = 5/2(10 + 12) = 55. Hence option B is correct. Starting with n = 0 would produce a different list and is not appropriate when the first term convention is n = 1. Other distractors can result from omitting a term or making an addition error.

Correct answer: B. (3000)

Here (a=200), (d=50), and (n=8), so the total is (3000) rupees. In money questions, keep the unit in mind.

Correct answer: D. (572)

The thirteenth term is (86), so (S_{13}=\frac{13}{2}(2+86)=572). Use ((n-1)d) when finding the last term.

Correct answer: B. 177

The governing concept is the meaning of partial sums. S_11 is the sum of terms 1 through 11, whereas S_5 is the sum of terms 1 through 5. Subtracting S_5 from S_11 cancels the common first five terms and leaves exactly terms 6, 7, 8, 9, 10, and 11. Therefore, the required sum is S_11 − S_5 = 242 − 65 = 177. Hence option B is correct. This method is valid for any sequence, and the fact that the sequence is an AP is not needed for this particular subtraction. Options A, C, and D may arise from an arithmetic subtraction error or from accidentally including the fifth term or excluding the eleventh term. No first term or common difference needs to be calculated.

Correct answer: B. (306)

The sum of the first (n) even numbers is (n(n+1)), so (17\times18=306). Do not confuse (n) with the last even number.

Correct answer: B. (315)

The seventh term is (30), so (S_7=\frac{7}{2}(60+30)=315). The same sum formula works for a decreasing progression.

Correct answer: C. 21

The formula for average is \(\text{average}=\frac{\text{sum of terms}}{\text{number of terms}}\). Hence, \(\frac{126}{6}=21\), so 21 is correct. Values such as 20 or 22 result from not dividing the total sum correctly by 6. Exam tip: for an average question, divide the total sum by the number of terms first.

Correct answer: A. (808)

The sixteenth term is (88), so (S_{16}=\frac{16}{2}(13+88)=808). For larger (n), find the last term first.