Forever Squared — Print Cards

∞ The Scenario

Background — your teacher will read this aloud

A designer starts with a single square. At every stage, new squares are attached to every free edge of the previous generation — each one half the side length of the square it touches. The pattern grows forever. Your job: figure out how much area it covers, and what happens if it never stops.

Phase 1

The Pattern

Task 1a — Build the table

Complete this on your whiteboard for stages 0–4:

Stage	New squares	Side	Area each	New area
0	1	1	1	1
1	?	?	?	?
2	?	?	?	?
3	?	?	?	?
4	?	?	?	?

Task 1b

What is the common ratio \(r\) of the new area column (from Stage 1)?

Task 1c

Write a general expression \(a_n\) for the new area added at stage \(n\). Find \(a_{10}\).

Phase 2

The Total Area

Task 2a

Find the total area after stages 1, 2, and 3 by adding terms.

Task 2b

Total area after 10 stages — find a shortcut.

Write: \(S = u_1 + u_1 r + u_1 r^2 + \cdots + u_1 r^9\)
Below: \(rS = u_1 r + u_1 r^2 + \cdots + u_1 r^{10}\)
Subtract. What cancels?

Task 2c

Derive a general formula for total area after \(n\) stages.

Phase 3

Forever

Task 3a

Use your formula to find the total area after 5, 10, 20, and 50 stages. What do you notice?

Task 3b

What happens to \(r^n\) as \(n \to \infty\)? Write a formula for \(S_\infty\).

\(S_\infty\) only exists when \(|r| < 1\).

Task 3c

Calculate \(S_\infty\) for this fractal. What is the total area if the pattern grows forever?

Phase 4

The Other Side of Infinity

Task 4a

Look at the count column: 1, 4, 12, 36, … Is this geometric? What is its ratio?

Task 4b

Does the total number of squares converge as \(n \to \infty\)? Use the \(|r| < 1\) condition.

Task 4c — The paradox

Infinitely many squares, yet total area = 5. Write one sentence on your board explaining why this isn't a contradiction.

HL Extension

A fractal starts with area 1 and multiplies new area by \(r\) each stage. For what values of \(r\) does total area converge? Find \(r\) that gives total area = 3.