Word problems based on APs

The drawing numbers are (55,66,77,\ldots) and (a_{21}=275). Exam tip: work on the (n)th day is found by the (n)th term.

The page numbers are (130,147,164,\ldots) and (S_{14}=3367). Exam tip: use the sum formula for total printing.

The practice times are (45,60,75,\ldots) and (S_n=2520) gives (n=16). Exam tip: set the target total equal to (S_n).

Using (S_{20}=2150) and (d=5) gives (a=60). Exam tip: find the first day from total and increase.

The packets form the decreasing AP (300,290,280,\ldots) and (S_{18}=3870). Exam tip: keep the common difference negative for decreasing quantities.

The number of seats forms an arithmetic progression with first term \(a=40\) and common difference \(d=7\). The number of seats in the \(n\)th row is \(a_n=40+(n-1)\times 7\). Setting this equal to 187 gives \(40+(n-1)\times 7=187\), so \(7(n-1)=147\), \(n-1=21\), and \(n=22\). Thus, the 22nd row has 187 seats. The 24th row would have \(40+23\times7=201\) seats, so it is not correct. Exam tip: in row-number AP questions, equate the given value to \(a_n=a+(n-1)d\).

The seats form an AP and (S_n=2520) gives (n=28). Exam tip: when total seats are given, form an (S_n) equation.

The salaries form the AP (42000,44500,47000,\ldots) and (S_{10}=532500). Exam tip: add all monthly salaries using the sum formula.

The production is the decreasing AP (600,578,556,\ldots) and (S_{18}=7434). Exam tip: treat a decrease as a negative common difference.

Using (S_{20}=1240) and (a=24) gives (d=4). Exam tip: find the common difference from the total and first term.

This forms an arithmetic progression with first term \(a=18\) and common difference \(d=9\). The number of plants in the \(n\)th row is \(a_n=18+(n-1)\times 9\). Putting \(a_n=207\), \(18+(n-1)\times 9=207\). Thus, \((n-1)\times 9=189\), so \(n-1=21\) and \(n=22\). Hence, the 22nd row has 207 plants. The 23rd row would have 216 plants, so it is not correct. Exam tip: equate the target quantity directly to \(a_n\) and solve for \(n\).

The daily work forms the AP (75,88,101,\ldots) and (S_{16}=2760). Exam tip: treat the daily increase as the common difference.

The donation amounts are (250,375,500,\ldots) and (S_n=9625) gives (n=11). Exam tip: form an (S_n) equation for the target total.

From \(180+(n-1)(-8)=68\), \(n=15\). Exam tip: handle the sign carefully in a decreasing AP.

The fares are (120,155,190,\ldots) and (S_{12}=3750). Exam tip: treat each kilometre's fare as one term.

The matchstick numbers are (22,33,44,\ldots) and (a_{19}=220). Exam tip: take the figure number as the term number.

The deposits are (450,510,570,\ldots) and (S_{15}=13050). Exam tip: treat weekly deposits as AP terms.

The amounts form the decreasing AP (1800,1725,1650,\ldots) and (S_{14}=18375). Exam tip: keep (d) negative for decreasing quantities.

The salary forms an AP and (a_{11}=53000). Exam tip: find a particular month's salary using the (n)th term.

The capacities are (48,56,64,\ldots) and (S_{18}=2088). Exam tip: add all rows for total capacity.