Word problems based on APs

Correct answer: C. (67)

The tile numbers are (7,12,17,\ldots) and (a_{13}=67). Exam tip: take the figure number as the term number.

Correct answer: C. (9)

Using (S_6=405) and (a=45) gives (d=9). Exam tip: daily increase can be found from total distance.

Correct answer: B. (484)

The packet numbers are (24,28,32,\ldots) and (S_{11}=484). Exam tip: take the number of rows as the number of terms.

Correct answer: A. (460)

The sales form the decreasing AP (75,70,65,\ldots) and (S_8=460). Exam tip: take (d) negative for decreasing sales.

Correct answer: B. (216)

The work hours are (12,15,18,\ldots) and (S_9=216). Exam tip: add all days using the sum formula for total hours.

Correct answer: B. (340)

The member numbers are (25,30,35,\ldots) and (S_8=340). Exam tip: treat monthly new members as AP terms.

Correct answer: C. (75)

The matchstick numbers are (9,15,21,\ldots) and (a_{12}=75). Exam tip: in pattern questions check whether the difference is constant.

Correct answer: C. (170)

The page numbers are (30,50,70,\ldots) and (a_8=170). Exam tip: find the (n)th term for a particular month.

Correct answer: A. (504)

Direct answer: Option A, 504 lights. The numbers form an arithmetic progression because every gate has 7 more lights than the previous gate. Here a=28, d=7, and n=9. First find the ninth term: \\(a_9=a+(n-1)d=28+8(7)=84\\). Then use the sum formula: \\(S_9=\\frac n2(a+a_9)=\\frac92(28+84)=\\frac92(112)=504\\). Option A is correct. Option B, 512, is too large and does not result from the formula. Option C, 520, is also not the calculated total. Option D, 528, is another incorrect total; it may come from adding terms incorrectly. The key is to add all nine gate counts, not multiply the last count by nine. Memory cue: AP sum = number of terms × average of first and last.

Correct answer: B. (140)

Using (S_{12}=3000) and (d=20) gives (a=140). Exam tip: the first term can be found from total sum and difference.

Correct answer: B. (540)

The patient numbers are (36,40,44,\ldots) and (S_{10}=540). Exam tip: use the sum formula for total patients.

Correct answer: C. (54)

The passenger numbers are (12,15,18,\ldots) and (a_{15}=54). Exam tip: take the stop number as the term number.

Correct answer: B. (6)

Using (S_{14}=721) and (a=12) gives (d=6). Exam tip: find the common difference from the total and first term.

Correct answer: A. (648)

The box numbers are (32,36,40,\ldots) and (S_{12}=648). Exam tip: total can also be found using the average of first and last shelves.

Correct answer: B. (400)

The user numbers are (150,175,200,\ldots) and (a_{11}=400). Exam tip: find a particular day's value using the (n)th term.

Correct answer: C. (8)

Using (S_9=450) and (a=18) gives (d=8). Exam tip: find the increase from total members and the first month.

Correct answer: B. (520)

The capacity forms the AP (16,20,24,\ldots) and (S_{13}=520). Exam tip: use the sum formula for total capacity of rows.

Correct answer: B. 720

This is an AP with \(a=20\), \(d=4\), and \(n=15\). \(S_{15}=\frac{15}{2}[2(20)+14(4)]=\frac{15}{2}(96)=720\). Finding only the 15th-row seats would not give the total. In exams, remember to use \(n-1\) in the sum formula.

Correct answer: C. (425)

The pot sales are (20,25,30,\ldots) and (S_{10}=425). Exam tip: use the sum formula to add sales across all days.

Correct answer: B. (700) किलोमीटर

The distances form the AP (70,80,90,\ldots) and (S_7=700). Exam tip: treat the distance of each hour as a separate term.

Correct answer: C. (58)

The number of drawings is (6,10,14,\ldots) and (a_{14}=58). Exam tip: work done on the (n)th day is the (n)th term.

Correct answer: B. (210)

The sales form the decreasing AP (45,40,35,\ldots) and (S_7=210). Exam tip: in decreasing sales, (d) is negative.

Correct answer: C. (8)

The direct answer is option C, an increase of 8 sacks per rack. The rack counts form an arithmetic progression with first term \(a=20\), number of terms \(n=8\), total \(S_8=384\), and common difference \(d\). Use \(S_n=\frac n2[2a+(n-1)d]\). Thus \(384=\frac82[40+7d]=4(40+7d)\). Dividing by 4 gives \(96=40+7d\), so \(7d=56\) and \(d=8\). Option A, 6, would give a total of 352, not 384. Option B, 7, would give 380. Option C, 8, gives the stated total and is correct. Option D, 9, would give 408. The important step is to use the total sum, not only the last rack.

Correct answer: A. (242)

The number of computers is (7,10,13,\ldots) and (S_{11}=242). Exam tip: use the sum formula for total installation.

Correct answer: B. 31 जीबी

The governing concept is the nth term of an arithmetic progression. The daily amounts are 3, 5, 7, and so on, so the first term is a = 3, the common difference is d = 2, and the required term is n = 15. Using aₙ = a + (n − 1)d, we get a₁₅ = 3 + (15 − 1) × 2 = 3 + 28 = 31 GB. Hence option B is correct. The question asks for the amount on one particular day, so summing all fifteen days would be inappropriate. Option A uses too little increase, while 33 and 35 use excess increase or treat the first day as an additional step. Counting fourteen increments after day one is the essential point.