Exam Hall Seating Plan
St. Xavier's Higher Secondary is converting its old assembly hall into an examination centre. The carpenter's Plan A puts 24 seats in the front row and adds 4 seats to every row behind it. The sloped ceiling allows at most 18 rows, and no row may contain more than 100 seats because of aisle clearance. The examination board requires at least 950 usable seats. A second contractor, Plan B, proposes 30 seats in the front row with an increase of 2 seats per row, also within the same 18-row limit. The school must choose the plan that reaches the 950-seat requirement with the fewest rows, without violating the 100-seat row cap. The principal also wants to know the total capacity if all 18 rows are built under the chosen plan, so that the extra seats can be offered to a neighbouring school.
1. For Plan A, write an expression for the number of seats in row n and calculate the total capacity if all 18 rows are built.
Read the worked answer
Plan A is an arithmetic sequence with first term $a=24$ and common difference $d=4$.
Row $n$ has
$$a_n=24+(n-1)4=4n+20.$$
So row 18 has $4(18)+20=92$ seats, which is within the 100-seat cap.
The total for 18 rows is
$$S_{18}=\frac{18}{2}(24+92)=9\times116=1044.$$
Thus Plan A can provide 1044 seats if all 18 rows are built.
2. Find the minimum number of rows under Plan A needed to reach 950 seats, and verify that the last required row does not break the 100-seat cap.
Read the worked answer
For Plan A,
$$S_n=\frac{n}{2}[2(24)+(n-1)4]=\frac{n}{2}(4n+44)=2n^2+22n.$$
Check near the target:
$S_{16}=2(256)+22(16)=512+352=864$, too small.
$S_{17}=2(289)+22(17)=578+374=952$, enough.
So the minimum is 17 rows. The 17th row has $4(17)+20=88$ seats, which is below 100. The school should build 17 rows, giving 952 usable seats.
3. Evaluate Plan B and recommend which plan the school should adopt.
Read the worked answer
Plan B has $a=30$ and $d=2$, so
$$S_n=\frac{n}{2}[60+(n-1)2]=n^2+29n.$$
Even at the maximum 18 rows,
$$S_{18}=18^2+29(18)=324+522=846,$$
which is below 950. Plan B cannot meet the board requirement within the ceiling limit.
Plan A reaches 952 seats with 17 rows and respects the 100-seat row cap. Therefore the school should adopt Plan A with 17 rows; if all 18 rows are built, the capacity is 1044 seats.