Word problems based on APs

Putting (S_n=2328) in the sum formula gives (n=24). Exam tip: when total seats are given, form an equation for (n).

The salaries form the AP (35000,36800,38600,\ldots), and (S_9=379800). Exam tip: add all monthly salaries using the sum formula.

The production is the decreasing AP (420,402,384,\ldots) and (S_{15}=4410). Exam tip: treat a decrease as a negative common difference.

For the first 20 AP terms, \(S_n=\frac{n}{2}[2a+(n-1)d]\). Here \(n-1=19\), so \(S_{20}=10[50+57]=1070\). Twenty rows create only 19 increases. Exam tip: check the \((n-1)d\) term carefully.

The amounts are the AP (90,120,150,\ldots) and (S_{14}=3990). Exam tip: treat the number of days as the number of terms.

The packet numbers form a decreasing AP and (S_{14}=4725). Exam tip: take a decrease as a negative common difference.

Using (S_{16}=1320) and (a=45) gives (d=5). Exam tip: find the common difference from total and first term.

The numbers of flowers form an AP with first term \(a=42\) and common difference \(d=11\). The number of flowers in the \(n\)th layer is \(a_n=42+(n-1)\times 11\). Putting \(a_n=207\), \(42+(n-1)\times 11=207\), so \((n-1)\times 11=165\), giving \(n-1=15\) and \(n=16\). Hence, the 16th layer has 207 flowers. The 15th layer would have only \(196\) flowers. Exam tip: when a position or layer is asked, equate the target value to \(a_n\) and solve for \(n\).

The production is the AP (70,83,96,\ldots) and (S_{12}=1698). Exam tip: treat each hour's production as one term.

Using (S_{18}=1485) and (d=5) gives (a=40). Exam tip: solve the sum formula backward to find the first term.

The amount of water forms a decreasing AP with \(a=210\) and \(d=-12\). On the \(n\)th day, \(a_n=210+(n-1)(-12)\). Solving \(210-12(n-1)=66\) gives \(n-1=12\), so \(n=13\). Therefore, 66 litres remain on the 13th day. On the 12th day, the amount is 78 litres, so option 12 is not correct. Exam tip: use a negative common difference \(d\) for a decreasing AP.

The fares form the AP (85,115,145,\ldots) and (S_{11}=2585). Exam tip: treat each kilometre's fare as one term.

Using (S_{20}=1230) and (a=14) gives (d=5). Exam tip: daily increase can be found from total practice.

The coin numbers are (75,110,145,\ldots) and (S_{16}=5400). Exam tip: treat level rewards as an AP.

The weekly sales form an AP with first term \(a=95\) and common difference \(d=18\). The sale in the \(n\)th week is \(a_n=95+(n-1)18\). Setting this equal to 311 gives \(95+(n-1)18=311\), so \(18(n-1)=216\), \(n-1=12\), and \(n=13\). Therefore, 311 kg will be sold in the 13th week. In the 12th week, the sale would be \(293\) kg, so it is a close but incorrect option. Exam tip: equate the target quantity to \(a_n\) and solve for \(n\).

The lights form the decreasing AP (96,92,88,\ldots) and (S_{18}=1116). Exam tip: in decreasing rows the common difference is negative.

The distances form the AP (28,35,42,\ldots) and (S_{15}=1155). Exam tip: treat each hour's distance as one term.

Using (S_{14}=2590) and (d=12) gives (a=107). Exam tip: find the first day from total and increase.

The amounts are (60,80,100,\ldots) and (S_n=2040) gives (n=12). Exam tip: put the target total in the sum formula.

The salaries are (18000,20200,22400,\ldots) and (S_7=172200). Exam tip: add salary increases like an AP.

The box numbers are (320,300,280,\ldots) and (S_{11}=2420). Exam tip: take (d) negative for decreasing stock.

The points are (120,165,210,\ldots) and (a_{18}=885). Exam tip: take the level number as the term number.

The units form the AP (150,175,200,\ldots) and (S_{10}=2625). Exam tip: add consumption across all months.

The number of windows forms a decreasing AP with \(a=150\) and \(d=-7\). Thus, \(a_n=150+(n-1)(-7)\). Substituting \(a_n=52\) gives \(150-7(n-1)=52\), so \(7(n-1)=98\) and \(n=15\). The 14th floor would have 59 windows and the 16th floor would have 45, so the correct answer is the 15th floor. Exam tip: write the common difference as negative for a decreasing AP.

The instalments are (7500,8750,10000,\ldots) and (S_6=63750). Exam tip: treat instalments as AP terms.