Word problems based on APs

The pot numbers are (18,25,32,\ldots) and (S_{16}=1128). Exam tip: use the sum formula for total production.

The admissions are (45,60,75,\ldots) and (S_n=2520) gives (n=16). Exam tip: connect the target total with the sum formula.

This is an arithmetic progression because the production increases by the same number, 15 units, every hour. The first term is \\(a=95\\), the common difference is \\(d=15\\), and the fifteenth hour represents the fifteenth term. The formula is \\(a_n=a+(n-1)d\\). Substituting the values gives \\(a_{15}=95+(15-1)\\times15=95+210=305\\). Thus the machine produces 305 units in the fifteenth hour.

Option B is correct. A common mistake is to multiply 15 by 15 without accounting for the first hour, or to use 15 increases instead of the 14 increases that occur before the fifteenth hour. The sequence begins 95, 110, 125, and continues by adding 15. Since reaching the fifteenth term requires fourteen additions, 305 is the required production.

The work hours are (20,25,30,\ldots) and (S_{14}=735). Exam tip: add all days using the sum formula for total hours.

The instalments are (6000,7000,8000,\ldots) and (S_7=63000). Exam tip: treat instalments as AP terms.

Using (S_{18}=1197) and (d=5) gives (a=24). Exam tip: find the first term from total and difference.

The sales form the decreasing AP (220,206,192,\ldots) and (S_{12}=1716). Exam tip: keep (d) negative for decreasing sales.

This is an arithmetic progression with first term 40 and common difference 20. The number of pages read in the nth month is \(a_n=40+(n-1)\times20\). For 300 pages, \(40+(n-1)\times20=300\). Thus, \((n-1)\times20=260\), so \(n-1=13\) and \(n=14\). Therefore, the student will read 300 pages in the 14th month. In the 13th month, the student would read only 280 pages. Exam tip: use the nth-term formula here, not the formula for the sum of terms.

The distances are (25,30,35,\ldots) and (S_{18}=1215). Exam tip: take the number of gaps as the number of terms.

The member numbers are (30,38,46,\ldots) and (S_{15}=1290). Exam tip: treat monthly new members as AP terms.

The passenger numbers are (12,19,26,\ldots) and (S_{16}=1032). Exam tip: take the number of stops as the number of terms.

The profits form the AP (75000,84000,93000,\ldots) and (S_7=714000). Exam tip: treat yearly profits as separate terms.

The tile numbers are (12,20,28,\ldots) and (S_{18}=1440). Exam tip: find the total of a pattern using the sum formula.

The sales form the AP (150,170,190,\ldots) and (S_{10}=2400). Exam tip: treat daily sales as AP terms.

Using (S_{15}=8550) and (a=150) gives (d=60). Exam tip: find the common difference from total and first term.

The amount filled each minute forms an arithmetic progression with first term 30 and common difference 11. Thus, the amount in the nth minute is 30 + (n-1)×11. For 206 litres, 30 + (n-1)×11 = 206, so (n-1)×11 = 176 and n = 17. Therefore, 206 litres are filled in the 17th minute. In the 16th minute, the amount is 195 litres, so it is a close but incorrect option. Exam tip: use a_n = a+(n-1)d when a particular term of an AP is given.

The patient numbers are (40,48,56,\ldots) and (S_{18}=1944). Exam tip: use the sum formula for total patients.

The sack numbers are (60,70,80,\ldots) and (S_{11}=1210). Exam tip: treat each row as one term of the AP.

The video numbers are (8,13,18,\ldots) and (S_{22}=1331). Exam tip: take the number of weeks as the number of terms.

The lane numbers are (9,13,17,\ldots) and (S_{19}=855). Exam tip: treat a sequence with equal increase as an AP.