Word problems based on APs

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

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

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.

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.

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

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

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.

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

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.

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.

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

Using (S_8=384) and (a=20) gives (d=8). Exam tip: find the common difference from the total and first term.

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

The daily data amounts form an arithmetic progression because the same amount, 2 GB, is added each day. Here the first term is a = 3, the common difference is d = 2, and the required day is the 15th term. Use the nth-term formula aₙ = a + (n − 1)d. Therefore, a₁₅ = 3 + (15 − 1) × 2 = 3 + 28 = 31 GB. Hence option B is correct. Option A results from using only 14 GB of increase incorrectly, while 33 and 35 arise from adding too much or treating the first day as an additional increase.

The cloth sales form (60,70,80,\ldots) and (S_9=900). Exam tip: use the sum formula for sales across all days.

This forms an arithmetic progression with first term 9 and common difference 4. The number of instruments on the nth day is 9 + (n - 1) × 4. Setting it equal to 57 gives 9 + (n - 1) × 4 = 57, so 4(n - 1) = 48, n - 1 = 12, and n = 13. Therefore, 57 instruments will be placed on the 13th day. On the 12th day, there would be only 53 instruments. Exam tip: in AP word problems, first identify the first term and common difference, then use the nth-term formula.

The practice times form the AP (35,40,45,\ldots) and (S_{16}=1160). Exam tip: use the sum formula for total time.

The seats form an AP and (a_{17}=25+16(4)=89). Exam tip: treat the row number as the term number.

The user numbers are (200,240,280,\ldots) and (S_9=3240). Exam tip: treat the daily increase as the common difference.

Putting (S_n=425) in the sum formula gives (n=10). Exam tip: when total sum is given, form an equation for (n).