Stand Up & Math · Activity
Forever Squared
Geometric sequences & series · Groups of 3–4 · Whiteboard
A designer starts with a square of side length 1. At each stage, a new square — with side length half the previous — is attached to the centre of every free edge of the squares added in the previous stage. The first four stages look like this:
For stages 0 to 4, complete this table on your whiteboard:
| Stage \(n\) | New squares added | Side length of each | Area of each | Total new area added |
|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 |
| 1 | ? | ? | ? | ? |
| 2 | ? | ? | ? | ? |
| 3 | ? | ? | ? | ? |
| 4 | ? | ? | ? | ? |
Look at the "Total new area added" column. What is the ratio between consecutive terms? Is it constant?
This type of sequence — where each term is found by multiplying the previous by a fixed number — is called a geometric sequence. That fixed number is called the common ratio \(r\).
Let \(a_n\) be the total new area added at stage \(n\). Write a general expression for \(a_n\) in terms of \(n\). Use it to find \(a_{10}\) — the new area added at stage 10.
Find the total area of the fractal after stages 1, 2, and 3. Add the terms by hand.
You're adding the new area from each stage to the previous total.
Now find the total area after 10 stages. Is there a faster way than adding 11 terms one by one?
Try this. Write the sum as:
\(S = u_1 + u_1 r + u_1 r^2 + \cdots + u_1 r^{9}\)
Now write a second line directly below it, where every term is multiplied by \(r\):
\(rS = u_1 r + u_1 r^2 + u_1 r^3 + \cdots + u_1 r^{10}\)
Look at the two lines side by side. What do you notice? What happens if you subtract the second from the first?
Use the pattern from 2b to derive a formula for the total area after \(n\) stages. Write it in terms of \(a_0\), \(r\), and \(n\).
Using your formula from Phase 2, calculate the total area after 5, 10, 20, and 50 stages. What do you notice? Does the total keep growing forever, or does it approach a fixed value?
Look at your formula for \(S_n\). What happens to \(r^{n}\) as \(n \to \infty\)? Use this to write down a formula for the total area if the process continued forever.
This is called the sum to infinity, written \(S_\infty\). It only exists when \(|r| < 1\).
Calculate \(S_\infty\) for this fractal. Is the answer what you expected? Could you have guessed it from the diagram?
Go back to your table from Phase 1. Look at the "new squares added" column — not the areas, just the counts: 1, 4, 12, 36, …
Is this also a geometric sequence? If so, what is its common ratio?
Write the total number of squares after \(n\) stages as a sum. Does this sum converge to a finite number as \(n \to \infty\), or does it grow without bound?
Use the condition \(|r| < 1\) from Phase 3. What happens here?
So: the fractal has infinitely many squares — yet their total area is exactly 4. How is this possible? Discuss with your group and write a sentence on your board that explains why this isn't a contradiction.
A different fractal starts with a square of area 1 and at each stage adds new squares with total new area multiplied by \(r\). For what values of \(r\) does the total area converge? Find a general formula for \(S_\infty\) in terms of \(r\), and determine which value of \(r\) gives a total area of exactly 3.
You've worked through all four phases of Forever Squared.
Well done — get ready for the class consolidation.