Exploding Dots
1.9 An Infinite Process: The Geometric Series Formula
Lesson materials located below the video overview.
Consider again an \(1 \leftarrow x\) base machine.
We can use this machine to divide \(1\) by \(1x\), that is, to compute \(\dfrac{1}{1x}\).
The quantity “\(1\)” is a single dot in the units position and the quantity “\(1x\)” is an antidot in the \(x\) position.
We wish to find copies of in the picture . Of course there are none at this stage.
The trick is to fill and empty box with a dot and antidot pair. This gives us a copy of in the units position.
We can repeat this trick:
and again, infinitely often!
This shows that, as a statement of algebra, we have:
\(\dfrac{1}{1x}=1 + x + x^{2} + x^{3} + …\)
Question 45:
a) Use this technique to show that \(\dfrac{1}{1+x} = 1 – x + x^{2} – x^{3} – x^{4} + …\).
b) Compute \(\dfrac{x}{1x^{2}}\).
c) Compute \(\dfrac{1}{1 – x – x^{2}}\) and discover the Fibonacci numbers!

Even more fun thoughts about the geometric series appears in the COMPANION GUIDE to this EXPLODING DOTS course – as well as some commentary as to when the formula is true as a statement of arithmetic, not just a statement of algebra.
Resources
Books
Take your understanding to the next level with easy to understand books by James Tanton.
BROWSE BOOKS
Guides & Solutions
Dive deeper into key topics through detailed, easy to follow guides and solution sets.
BROWSE GUIDES
Donations
Consider supporting G'Day Math! with a donation, of any amount.
Your support is so much appreciated and enables the continued creation of great course content. Thanks!
Ready to Help?
Donations can be made via PayPal and major credit cards. A PayPal account is not required. Many thanks!
DONATE