The Auditorium

IBDP Mathematics AA · SL & HL · 1.2 · Arithmetic sequences & series
🎭 The Scenario

Background — your teacher will read this aloud

A concert hall is being designed. The architect has one rule: every row must have a fixed number of extra seats compared to the row in front. You are the venue consultants. Your job is to answer the client's increasingly difficult questions — starting with the simplest design and working up to the full hall. All work goes on the whiteboard first.

1
Phase 1
Reading the Pattern
Context
The first four rows of one design:
Row 1: 12 · Row 2: 15 · Row 3: 18 · Row 4: 21
Task 1a
How many seats in Row 10? Row 20?
Work it out — don't use a formula.
Task 1b
How many seats in Row \(n\)?
Write a general expression for any row number \(n\).
Task 1c
A VIP row has exactly 63 seats. Which row is it?
2
Phase 2
A New Theatre
Context
A second design — only two rows are known:
Row 5: 38 seats  ·  Row 12: 59 seats
Task 2a
Find \(d\) and the number of seats in Row 1.
Task 2b
Write the general term \(u_n\).
Task 2c
Rows are built up to 150 seats max. How many rows does the theatre have?
3
Phase 3
The Sum Problem
Context
Third design: Row 1 has 20 seats, each row adds 6.
Task 3a
Total seats in the first 5 rows.
Use any method.
Task 3b
Total seats in the first 15 rows.
Task 3c
The hall has 40 rows. Total seats?
Can you think of a way that doesn't involve adding 40 different numbers?
Task 3d — Verify
30 rows · starts at 8 seats · adds 5 per row.
Find the total using your method from 3c.
4
Phase 4
Sigma Notation & Reverse Problems
Task 4 — Intro
From one formula to another.
You know \(S_n = \tfrac{n}{2}(u_1 + u_n)\). Replace \(u_n\) using the expression from Phase 1 and simplify to get a formula using only \(u_1\), \(d\), and \(n\).
Task 4a
Decode and compute.
The symbol \(\Sigma\) is the Greek capital letter sigma — the Greek S, used here because S stands for sum. With that in mind, figure out what this is asking and find its value: \[\sum_{k=1}^{25}(7 + 4(k-1))\] Then write the sum from Task 3c in the same notation.
Task 4b
Working backwards.
A theatre has 1 350 seats total across 25 rows, with \(d = 4\). Find \(u_1\).
HL Extension

A second theatre has \(d = 3\), its last row has 49 seats, and total seats = 424. Find \(n\) and \(u_1\).