A concert hall is being designed. The architect's rule: each row has a fixed number of extra seats compared to the row in front. The first four rows of one design look like this:
How many seats are in Row 10? How many in Row 20? Work it out — don't use a formula.
How many seats are in Row \(n\)? Write a general expression for any row number \(n\).
A VIP row has exactly 63 seats. Which row number is it?
The architect has a second design. This time you are not given the first row — only two data points:
Find the common difference \(d\) and the number of seats in Row 1.
Write the general term \(u_n\) for this theatre.
The theatre only builds rows with up to 150 seats. How many rows does it have?
A third theatre design: Row 1 has 20 seats, and each row adds 6 more than the one before.
Find the total number of seats in the first 5 rows. Use any method.
Find the total number of seats in the first 15 rows.
The client wants to know: the finished hall will have 40 rows. What is the total number of seats?
Can you think of a way that doesn't involve adding 40 different numbers?
A different theatre has 30 rows, starts at 8 seats in Row 1, and adds 5 seats per row. Use your method from 3c to find the total number of seats.
In Phase 3 you found the formula \(S_n = \dfrac{n}{2}(u_1 + u_n)\). This is useful when you know the first and last terms — but what if you only know \(u_1\), \(d\), and \(n\)?
Replace \(u_n\) in the formula above using the expression for \(u_n\) you found in Phase 1. Simplify until you get a formula for \(S_n\) that uses only \(u_1\), \(d\), and \(n\).
The symbol \(\Sigma\) is the Greek capital letter sigma — the Greek equivalent of the letter S, used here because S stands for sum. With that in mind, figure out what the following expression is asking you to compute, and find its value:
\(\displaystyle\sum_{k=1}^{25}(7 + 4(k-1))\)
Once you have the answer, write the sum from Phase 3, Task 3c in the same notation.
A theatre has a total of 1 350 seats spread across 25 rows, with a common difference of 4 seats per row. Find the number of seats in the first row.
A second theatre satisfies all of these conditions:
Find the number of rows \(n\) and the number of seats in the first row \(u_1\).
You've worked through all four phases of The Auditorium.
Well done — get ready for the class consolidation.